While I can't exactly speak for all medieval armies, I can give one, very specific but perhaps indicative example (as such don't expect this answer to be massively long!); Henry V's army for his Agincourt campaign.
I suppose the best place to start would be the position of a knight's servant, which while not necessarily constituting a logistical role, they still offered non-combat support during campaigns, be it in the form of organising and leading their master's baggage or simply serving him when in camp. During Henry V's campaign, most knights were permitted to take six horses at the king's expense (though could theoretically take more at his own); one or two of these six horses would be for his servant(s), indicating the number that we might have seen for non-combat servant roles; with every knight, there was at least one or two servants on campaign, though the wealthier and perhaps more important knights would likely have brought even more. To quote Juliet Barker, "even the humblest esquire, with his four horses, must have been expected to bring a warhorse [for battle], a palfrey [for travel], a rouncy [for a servant] and a packhorse [for baggage], a retinue which presupposes that he had at least one or two servants to look after them for him". While it's possible that a number of knightly servants were archers in the king's army, most will have been boys and non-combatants. Because these were non-combatants, they don't figure on the indentures or muster rolls we have from the campaign, but one might assume a rather large number of non-combatant servants during this campaign.
To further address non-combatant roles, an example we can find for the campaign is the wage accounts of earl John Mowbray for this campaign, which include wages paid at military rates to a small group of non-combatants; two heralds, three minstrels and a trumpeter (colourfully named Thomas Trumpet). While it cannot necessarily be said that similar numbers were taken up by every earl, it's somewhat indicative of the kinds of non-combatants we might find on campaign (that is; not necessarily support, but there nonetheless).
Chancery rolls for the campaign also give insight into a number of non-combatant roles, through the profession of applicants for protection during the campaign (so, those who would join but require protection, as they couldn't fight); we have rectors, notaries, and chaplains. Smiths as well were brought on campaign (this was common across this period as well), whose professional skills were considered more valuable than their fighting skills, so they likely would have remained non-combatants, instead working on repairs and maintenance of weapons; see, for example, the king's own smith, William Merssh; or John Persshall, a bladesmith from London in the retinue of one Nicholas Merbury. Grocers, fishmongers and merchants were also employed in a professional capacity during the campaign, acting as suppliers to the army's various retinues. Some of the more surprising professions to be found in the rolls also indicate that some people perhaps preferred luxury even on campaign; we have two tailors, a baker from Norwich, a Woolman from Coventry, a butcher from Petworth, a drover from London, and a taverner in the Duke of Clarence's retinue. While we might assume that these professionals were hired in a combat capacity, we also have letters of protection granted to men whose stated profession is 'archer', 'man-at-arms', 'swordsman' etc.; combat roles tended to be specified in the chancery rolls.
The king's personal company during the Agincourt campaign essentially worked like a moving household, with a number of non-combatants in the ranks; the Master of Horse brought with him sixty grooms, a surveyor and clerk for the stable, a clerk and twelve yeoman purveyors of oats, twelve smiths, nine saddlers and two men whose role was simply to guide the king at night; there were also six bowyers, six fletchers, a helmet specialist and twelve armourers were employed to tend to the king's weapons and armour; while John Conyn, the king's sergeant of tents and pavilions, employed four painters and twenty-eight servants to look after the royal pavilions. That's a pretty large number of non-combatants to be found in the king's household; but, of course, being the king, it is likely his retinue outnumbers many others besides within his army.
Similar numbers, though, can be found in the king's kitchen staff, which were also brought on campaign; the king's kitchen supervisor employed three yeoman and a clerk of poultry, eight yeoman and a clerk of his bakehouse, three clerks of his spicery, a clerk of table-linen and another of his hall, a clerk and fifteen labourers for the scullery, and another 156 yeoman and servants assigned to no particular department. The king also had a number of carpenters, labourers, pages , clerks, almoners, treasurers and clerics who accompanied him on campaign, and (for the most part) would have worked in a non-combat capacity. It's worth noting that of the clerics we see a rather large number; a doctor of law, the dean of the king's chapel, three clerks, fifteen chaplains and fourteen monks. These men would have remained at the baggage train during the battle (and, indeed, one month wrote an account of the campaign, the Gesta Henrici Quinti, and talks of remaining at the baggage train praying for victory even as the battle of Agincourt raged around him).
The king also brought with him eighteen minstrels, and while similar numbers were unlikely to be found in the lesser nobility (the king was a music lover besides being, well, the king), it was equally as unlikely that any noble or aristocrat left for Agincourt with no minstrels whatsoever. Heralds also will have acted in a non-combat role, and would even sit aside with the heralds from the other side during a battle; indeed, it was often the heralds themselves who declared victory one way or the other, admitting defeat when it was obviously the case. Heralds acted as record-keepers and messengers, and wore specific badges of office (a tabard and a wand) to indicate his role and grant him diplomatic immunity even during war. Every noble had his own herald, while the king had at least five. Medical personnel were also numerous, and would have acted in a non-combat capacity. The king brought with him his own physician, as well as twenty-three surgeons (a number of whom were women).
There were a great many different non-combat roles to be found within the ranks of the English army during the Agincourt campaign; minstrels, clerics, heralds, butchers, bakers (possibly candlestick makers?), tailors, treasurers, bowyers, fletchers, smiths, servants, grooms and grocers. While it might be rash to assume that all of these professionals would be employed in a non-martial capacity, it might be equally rash to assume that all would have fought also, especially considered the value placed in many of these professions; smiths, for example, would likely not have fought, as their trade was difficult but valuable. Of course, it also goes without saying that there would have been a number of non-specified non-combatants on campaign, including women and boys acting in a servile capacity to tend to baggage, cook meals for the army, mend clothes and tend to horses. It's unfortunate, really, that most records and rolls from the campaign focus on the martial side of things, as it means we can only guess at the true number of non-combatants on campaign; but given the sheer size of the king's household (of non-combatants, given above), we can assume there was a fairly big bloody number!
Of course, Agincourt may well have been exceptional, and in no way indicative of wider trends in army composition, but it can still give a rough idea of what sort of roles we might be seeing, at least, in a medieval European army. I do hope my rambling here gives at least a little bit of help to your question!
Sources: Juliet Barker, 2015, Agincourt: The King, The Campaign, The Battle Anne Curry, 2015, Great Battles: Agincourt Anne Curry, 2002, The Hundred Years' War, 1337 - 1453
EDIT: I've just found another quick thing of non-combatants in the king's employ; Henry retained 124 carpenters, 25 cordwainers, 6 wheelwrights and 120 simple labourers for the purpose of maintaining and repairing siege machines!
The Greek historian Herodotos claimed that the Persian army that invaded Greece in 480 BC counted over 2.6 million soldiers. This number is impossible; Herodotos himself already doubted that anyone could feed so many people on campaign. But his doubt came partly from the knowledge that soldiers by themselves don't make an army, and Xerxes would have brought many more people along who didn't fight. While he tells us he was able to calculate the number of soldiers with some precision, he couldn't establish how many there were in the baggage train. But what he did instead is revealing:
As for the service-train which followed them and the crews of the light grain-bearing vessels and all the other vessels besides which came by sea with the force, these I believe to have been not fewer but more than the fighting men. Suppose, however, that they were equal in number, neither more nor fewer. If they were equal to the fighting contingent, they made up as many tens of thousands as the others. The number, then, of those whom Xerxes son of Darius led as far as the Sepiad headland and Thermopylai was 5,283,220.
That is the number of Xerxes' whole force. No one, however, can say what the exact number of cooking women, and concubines, and eunuchs was, nor can one determine the number of the beasts of draught and burden, and the Indian dogs which accompanied the host; so many of them were there.
-- Hdt. 7.186-187
Herodotos' working estimate was that there was at least one male servant, attendant or sailor for every soldier in the invasion force, and so in order to establish the total, he simply took the number of soldiers and doubled it. He went on to acknowledge that there were in fact many more people present if you included the women and eunuchs, but he didn't hazard a guess as to their number. In other words, in Xerxes' army, logistics and support "staff" was assumed to outnumber the fighting men.
Where did this working estimate come from? Elsewhere in his work, Herodotos enumerates the Greek army that gathered for the decisive battle of Plataiai in 479 BC. He has apparently precise numbers for the hoplite contingents of each of the 30+ states that made up the alliance. But when it comes to the light-armed troops, he has no such data, and instead assumes that he can just double the numbers:
The light-armed from the rest of Lakedaimon and Greece were as one to every hoplite, and their number was 34,500.
-- Hdt. 9.29.2
The only exception were the Spartans, who brought along as many as 7 light-armed helots for each hoplite. Now, Herodotos stresses that these were all fighting men, and we must assume that they carried missile weapons and other light equipment; but their primary function, highlighted by the fact that there was one for each hoplite, was to act as personal attendants for the wealthier, higher-status heavy infantry. The fact that the light-armed troops play no discernible part in the actual battle suggests that they were considered far more important as attendants, cooks, water carriers, grooms, craftsmen and camp guards than as warriors. Again, Herodotos counts only the men, and if we include the no doubt large numbers of camp followers who were women (in addition to any opportunistic merchants and tradesmen), we may assume that the support train again far outnumbered the combat-ready core of the army.
Other sources show the same principle. In order to show how expensive the Athenian siege of Potidaia was, Thucydides points out that the 4,000 hoplites committed to the operation were each "drawing 2 drachmai a day, one for himself and one for his servant" (3.17.4). Apparently the Athenian system of military pay took it for granted that every hoplite would have at least one attendant with him on campaign, who did not fight, but cooked food, built shelter, mended equipment, and so on. These attendants could be enslaved people, hired staff, or even young family members of the hoplite; the point is that every attested number of fighting men had a "shadow" of support personnel at least as large. According to Xenophon, those of Sparta's allies who wished could send money instead of men when the call to war came, at a rate of 3 Aiginetan obols per hoplite or 12 per horseman (Hellenika 5.2.21) - the difference reflecting the fact that cavalry would bring even more personal attendants per man and would have even more mouths to feed.
And once again, these are just the ones that are publicly accounted for. Hoplites and horsemen could bring many more people to attend and entertain them if they didn't mind paying out of their own pocket. In addition, every Greek army usually attracted travelling merchants who sold food and equipment to the soldiers, as well as their families, and anyone else who thought they might gain something from a large throng of bored, lonely men with drachmai in their pockets. It's not surprising that Greek armies tended to move slowly, and that they were often extremely concerned to protect their baggage train when on the march. On longer campaigns, these might even contain the lovers, wives and children of the fighting men.
To cut back on all this human and moral weight, Philip II of Macedon decreed that his soldiers could henceforth only have one servant for every 10 men, and had to carry their own equipment; this celebrated reform was one of the reasons why the Macedonian army of his reign and the early reign of his son Alexander could move with shocking speed compared to its regional enemies. But as these soldiers triumphed and their possessions and human following multiplied, the pendulum swung the other way again, and huge throngs of camp followers serving in support roles largely remained the norm throughout Antiquity.
As an addendum to u/doylethedoyle's excellent answer on the Agincourt campaign, and since the question specifically pertains to how many soldiers were used in logistics and support roles, it's important to look at the functioning of the levied troops as well - the common soldier, so to speak, rather than just the noble component and the non-military staff they brought with them.
I have written about the levy in high and late middle age Europe before, but suffice to say for now that those levied had to provide themselves with the equipment according to their wealth anon status, and had (broadly spoken) responsibilities that corresponded with the provided equipment and status. In 1342 Count Willem IV of Holland revised the laws concerning equipment, and those without any meaningful financial capabilities were expected to bring Brecbile, handbomen, ghetouwe ende andere gherieschap alsi si behoren om te woesten ende te vellen alle slote, stenhuze ende vestenisse dye onze vyanden hebben (crowbar, lever, rope and all other tools that one needs to take down the castles, reinforced houses and strongholds that are our enemies have) and to function as sappeurs; men that were so poor that they were not expected to bring any combat equipment whatsoever. Over time, the use of levied troops for support roles grew exponentially with the increasing employ of mercenaries and the subsequent "professionalization" of the army, to the degree that lords would rather have the levy default on their mustering and pay amercement instead (which would be used to hire mercenaries), or would bring the levy to bear solely for support roles, at times to the chagrin of aforementioned levy: in 1425, troops levied from the region of Kennemerland had to do so in quadruple levy (an intense form of levy that called for 3 (or 2.5 depending on the source) men where a normal levy would have to provide 1 man), leaving their settlements and farms relatively unattended, all just in order to do digging work, causing them to rise in rebellion.
Where it concerns logistics, it's more difficult to gain a picture of the role of the common levied soldier therein. The logistic apparatus used in the high and late middle ages is one far removed from the traditional image of it being crude and at times nonexistent, necessitating armies to roam and plunder the countryside instead (an image that does hold up to some extent in the early middle ages), which is a falsehood in most cases (these actions have to be seen primarely in the light of inhibiting the enemy's logistics). Exemptions are situations in which the normal logistical lines were inhibited, such as when a force is behind enemy lines (like Agincourt). Normally, however, medieval commanders were very much aware of how vital it was to provide and secure logistics for their armies, as becomes clear when one sees the many receipts of items being purchased in great quantities in the urban centres in preparation of a campaign. However, the receipts make clear that most of the tasks concerning large scale logistics seem to have been "outsourced" to non-levied non-military individuals with the means and tradecraft essential for logistics, such as wagoners, requisitioned ships (with crews!) and veerscutte (ferry men) to transport both troops and supplies, and carpenters to construct and employ the enghinen (siege engines). A dedicated logistics corps such as we know from the modern military was non-existent in this timeframe in this locale.
Sources
R. de Graaf (2004) - Oorlog om Holland (1000-1375)
Jansen & Hoppenbrouwers (1977) - Heervaart in Holland