It depended on the supply category.

For food and fodder, that depended on wagon trains, propositioned stores, contracted suppliers (ie bakers guilds) and carefully planned routes of march. Individual army commissaries would preceed the main force and either purchase or requisition the necessary food and fodder for the army. Logisticians of the time had actually worked out the supply potential of a given area of countryside and planned marches accordingly, when done mindfully the army did not have to cart much of its food and fodder in supply trains and could practically live hand to mouth on systematically requisitioned forage. In fact the need for what we consider supply lines and logistical trains only arose when corps marched on the same route in quick succession and exhausted the countryside, it is interesting to note that particular care was taken to prevent this from occurring because exhaustion of the countryside massively increased the logistical burden on the army. Even the practice of forced requisition of supplies was rarer than often thought, more often than not occurring only at points of extreme concentration of forces near the battlefield or along chokepoints along the march.

Martin Van Creveld states in Supplying War: Logistics from Wallenstein to Patton "As to the Grand Armée's ammunition supply, we saw that, in the nineteenth century, the latter exercised little or no influence on the strategic movements of armies owing to the very small quantities involved". A Napoleonic infantryman would generally not be expected to expend his personal allotment of ammunition during a battle, often let alone a campaign and in the case of the Austerlitz campaign a typical division of 8,000 men only brought along 97,000 infantry rounds in addition to the 60-80 carried by an infantryman. Ammunition expenditures for infantry remained low all the way through to the Franco-Prussian War and most ammunition demand was caused by guns (cannon) with Napoleon devoting 2500 of his 4500 wagons in the Austerlitz campaign to the artillery park. This seems significant but this was the planned expenditure for the entire campaign, there was little chance that any more than the initial contents of those ammunition wagons would be expended during the course of the entire campaign. Armies thus were unlikely to run out of ammunition and even if units were running low they likely had more than sufficient stocks on hand given the extremely low rate of expenditures normal for combat then.

Finally as in the case of the Austerlitz campaign, armies often replenished their supplies from those seized and found on the battlefield from fleeing enemies. Ammunition and arms were remarkably interchangeable between armies of the era and captured enemy depots such as at Heilbronn could supply enough arms and ammunition to supply a campaign many times over. Essentially ammunition resupply was never really a concern, especially when guts, determination and bayonets could readily route the most amply supplied opponent.

This post draws heavily on Martin Van Creveld's "Supplying War: Logistics from Wallenstein to Patton". It's an excellent read and I heartily recommend it.

I'm currently reading John Elting's Swords Around A Throne, and he mentions two solutions: One, that soldiers carried reserve ammunition in their packs; the other, that artillery caissons carried musket ammunition (and flints) in addition to their own ammunition. He mentions that divisional chiefs of staff were responsible for organising resupply, and that they would often assign a specific officer to carry this duty out. He doesn't mention precisely how ammunition got from the caissons to the men. (This is from page 491-2 in the paperback edition).