Arms dealing in the modern sense occurs when there is a technological gap between what the purchasers can buy locally and what is available from "foreign" sources who have desirable technologies that the locals cannot match. This creates a market opportunity for the foreign sellers, as well as political incentives for foreign statesmen to block those sales (in order to maintain a military advantage over the locals) and local statesmen to circumvent those controls (thus creating the conditions for a black market and criminal trading).
I am aware of one such case historically, and that is the arrival of matchlock firearms in Japan in the mid-16th Century. These were carried by Portuguese traders, who were quite happy to demonstrate their destructive power to the local Japanese, who in turn were smart enough to recognize that this was a game-changing military technology, and immediately began scheming to buy as much as they could.
The Lord of Tanegashima paid an exorbitant sum for the first guns to pass into Japanese hands, in the 1540s, and he immediately turned them over to his own weaponsmiths to figure out how to make more. However, the key technology was not the barrel or the firing mechanism, as you might think, but the threading of the breech. Japanese metal workers had no experience with that. So... back to the Portuguese for more arms dealing. This time the purchase was lessons in the fine points of gunsmithy from a Portuguese smith, and the price, according to legend, included the Lord's own daughter, which he willingly paid.
With that knowledge, the Lord of Tanegashima was able to produce something like 600 guns of his own, and turn around and start selling them in turn to others in Japan. With that, the Portuguese monopoly on matchlock gun technology waned, and within a generation Japan was fielding armies of gunners and winning battles with them. By the time flintlock weapons were developed in Europe in the early 17th Century, the Portuguese were restricted to trading through Nagasaki only. Flintlock weapons never took strong root in Japan, so for whatever reason, the black market incentives did not encourage much in the way of further arms dealing, despite the fact that flintlocks were a significant advance over matchlocks.
Arms and Armour of the Samurai, I Bottomley and A P Hopson, Chapter 6: The Arrival of the Southern Barbarians
I have a previous answer that goes through the entire arms and armour supply chain of later medieval armies, that touches on arms merchants. I have adapted some of the following answer from this one.
u/BlueStraggler has details about a case where foreign merchants supplied military hardware of a grade that is unavailable through local suppliers. However, another role of arms dealers is to provide -quantities- of arms that are not available locally, or provide them at a cheaper, bulk rate. In this case, we see a truly international and large scale arms trade within Europe as early as the High Middle Ages, if not earlier.* The reason for this is that while nearly every country had a domestic armoring industry, these were generally not sufficient to fill the 'rush orders' that would come in during war time, particularly when kings or great lords needed to buy armour and weapons for thousands of men. While by and large soldiers would supply their own equipment, and indeed, soldiers were drawn from those classes that could afford equipment (IE not the poorest tenant farmers and laborers), records show that kings supplemented the equipment to the tune of thousands of pieces of armour and tens of thousands of arrows and bows. Great lords would do likewise for their retinues. This created a market for bulk orders of armour and weapons. I will focus on armour because that is my greatest area of study.
There are roughly two classes of figures in the armouring industry that could be described as 'bulk arms merchants' - merchants proper, that bought from armourers or resellers and sold to purchasers of armour, and armourer-merchants, who controlled integrated supply chains enabling them to fill massive orders. The latter is a special case, and the best example of them is the Missaglia of Milan.
The earliest record I have seen of a truly massive arms purchase is 1295, when the merchant Frerderic the Lombard collected in Bruges the following items for the Fleet of King Philip of France:
2853 helmets
6309 round shields
4511 mail shirts
751 pairs of gauntlets
1374 gorgets
5067 coats of plates
This is a very large order for this early date. There are some smaller orders in the 14th and 15th centuries, in the low thousands. It is possible that there are some large orders whose records did not survive. In general, the price of armour and weapons declines as the Late Middle Ages progress into the Early Modern Period. By the 16th century, we see English Monarchs ordering infantry armours by the thousands, to the tune of as little as 7 shillings (less than half a pound) apiece. These armours or 'Almain Rivets' were cheaply constructed (of thin metal of indifferent quality) in Northern Germany, probably in the Wesphalia region, and then amassed into bulk orders by merchants in Koln, Antwerp or Amsterdam. The shops producing these armours may have been quite small - the scale was achieved by the merchant ordering from a large number of producers.
Regarding the Merchant/armourers, in Milan armourers were not restricted by the same guild rules that constrained the size of workshops in some German cities. Because of this one family, the Missaglia, were able to create a massive, vertically integrated operation that controlled all portions of the armour-making process from smelting ore to polishing the finished armour. They could subcontract individual pieces of armour to different workshops, and so construct even a full armour in as little as a day by cleverly dividing the labor. The Missaglia also had international operations in Naples, Barcelona and Tours. Alan Williams quotes the value of their operations at 100,000 scudi.** The armourers of Milan also seem to have kept quite a bit of armour on hand as well as being able to work fast. After the battle of Maclodio in 1427 the armourers of Milan were able to provide 4,000 cavalry armours and 2,000 infantry armours in a matter of days. I have not seen records of direct purchases from the Missaglia, however their operation was large and vertically integrated enough that it could fill massive orders. Unlike armour merchants in other places, the Missaglia controlled every step of the process and divided labor between subcontractors, creating an extremely efficient operation that achieved significant economies of scale. Their armour was also of famous quality - 15th century armour from Milan carrying an armouer's mark (from the Missaglia, their subcontractors, or someone else) is often heat treated to provide the best possible protection, and the steel is of good quality.
As a final note, a number of other aspects of the late medieval arms and armour industries required international trade on a massive scale. The yew for English longbows was, from the 14th century on, imported from Iberia or Italy in large quantities. Until Elizabeth I established nitre beds in England, English monarchs had to rely on foreign supplies for large orders of gunpowder, and they imported it in massive quantities during wartime. Medieval states did not possess extensive standing militaries or bureaucracies, so much of the logistics of medieval armies needed to be 'outsourced' to private individuals. Merchants of arms and armour were a critical part of this infrastructure.
Alan Williams - The Knight and the Blast Furnace
Thomas Richardson - The medieval inventories of the Tower armouries 1320–1410
Tobias Capwell - Armour of the English Knight 1400-1450
Edge and Paddock - Arms and Armour of the Medieval Knight
Strickland and Hardy - The Great Warbow
*I am not a student of the early Middle Ages or of antiquity, so I cannot speak to them.
**Williams does not provide an exchange rate between that and other currencies like the pound, but the scudo is the highest currency denomination of its system.
There is also the (probably at least semi-legendary) figure of Orban, selling his Cannon-casting skills to the Ottomans just before the fall of Constantinople in 1453. Supposedly, he offered his services in making extra-large siege cannons to the Byzantines first and after they didn't agree to his terms, he approached Mehmend II. who agreed.
He is mentioned in at least two sources: Leonardo di Chio and Nestor Iskander Further Reading: Runciman, Steven. The Conquest of Constantinople, 1453.
Perhaps one person you might be interested in is Francesco di Marco Datini of Prato. Apparently, he is the subject of a very good book called The Merchant of Prato by Iris Origo, but I know of him only from Cities by John Reader. While Datini and his partners engaged in many different kinds of trades, the book explains that he sold armor and weapons during his early career. The example given in the book:
...when a round of hostilities in Liguria ended in 1382, Francesco promptly instructed his agent in Genoa to buy up all the military equipment he could lay hands on, 'For, when peace is made, they are wont to sell all their armour,' and Francesco knew that sooner or later they, or some other party, would need it again.
I would say that fits the description of a private arms and armor dealer in the pre-modern era.
If you're interested in the early modern era of Europe, then absolutely yes. Military enterprises of all sizes and functions grew and developed as the nature of warfare changed. The rapid change from the high medieval model of mounted knights to Swiss pikes to pike & shot -- and the tercio system -- meant that almost everything had to change. From recruitment to training to the idea of a standing army. Of course, this meant that the weapons and methods of war changed rapidly.
As such, states large and small had to either change the way they run "the business of war" or perish. They had to be able to recruit, equip, train, and field larger armies than before. This meant that the 1500s-1600s saw the rise of mercenaries or perhaps more accurately military enterprisers.
The long and the short is this: European states initially had to rely on military enterprisers to run military operations and enable armies to be founded, run, and maintained. Over time, there were significant consolidation in the business, until the end of the Thirty Years' War saw the states themselves wrest control away from military enterprisers.
Back to your question, the model of military enterprises changed significantly over time. Italian city states prioneered the use of mercenaries in late medieval era. Swiss pikes rose to prominence because on one hand the Swiss cantons could not support the population that they ended up having due to extra income from working as mercenaries in the summers, something I wrote about here. But this ended up being their downfall in some cases. As they relied on plunder as a means to boost their income, they forced their French employers into a pitched battle at Bicocca when they did not get their demand for pay. Unfortunately, the Spanish Imperial troops had set up a defensive position that held against Swiss assaults.
In the 16th century there was no fluid, reliable banking and credit system that we know today, so in many endeavors even the wealthiest royals have to rely on private parties to put up with the upfront cost, in exchange for either nobility, land grant, or some other financial obligation. A large empire such as that of Charles V had to rely on enterprisers to simply get things done. Both he and his grandfather Emperor Maximilian I hired German landsknecht units, run similarly to Swiss pike units. In Italy, he hired Italian units to complement his core of Spanish troops.
When his son Philip II had to deal with Ottoman threats in the Mediterranean and the Dutch rebels, he too had to hire military enterprisers. In the Mediterranean he relied on the Doria family of Genoa, who ran galley fleets on behalf of the Spanish crown. When Philip tried to build his own galley fleet based off Catalonia and Seville, he found that it would have cost twice as much! The Taxis and Spinola families ran the link through Milan (a very important piece of Philip's empire in Europe) to Rhenish Palatine to the Low Countries, transporting goods, mail, and money.
When the Low Countries rebelled against his authority, he sent 9000 soldiers from Italy, under the Duke of Alba, up the so-called Spanish Road. This was a highly capitalized market, where Alba put up much of the cost up-front. But Alba did not have to recruit the entire Army of Flanders, as it was made of units already in Italy at that time. In that expedition, he contracted with many provisioners along the way, ensuring there were food and supplies throughout the journey. Yet once again, that army had a large component of mercenaries hired along the way and in the Low Countries itself.
His successor Philip III dealt with similar problems, and fortunately for him Ambrogio Spinola came with a bold proposal where he would manage the entire business of hiring an army, shipping it, fitting it, and leading it. This further integrated model perhaps found its pinnacle in the Thirty Years' War under Wallenstein. He did all Spinola did, but by having a much larger army, he gained permission from his erstwhile employer the Holy Roman Emperor to not only lead an army, but also set up a taxation system through which the army would be supported. To him it is attributed bellum se ipsum alet or war must feed itself, a phrase first coined by Cato the Elder.
In that same era, arms manufactures grew tremendously. They mostly operated in areas where there were happy meetings of the supplies of iron, coal, and men. These were cities such as Milan, Brescia, Augsburg, Leipzig, Liégé. Wallenstein's colonels made contracts to acquire armor from Milan and Brescia, muskets from Leipzig, and gunpowder from Austria and beyond! The Trip family of Amsterdam became involved in cannon manufacture for both the Swedish and French armies.
So the early modern era was a fascinating time that saw changes in the way armies were run and how arms were provisioned!
Sources:
J. Glete, "War and the state in early modern Europe: Spain, the Dutch Republic and Sweden as fiscal-military states, 1500-1600," ISBN-13: 978-0415226455, 2001.
F. Tallett, "War and Society in Early Modern Europe: 1495-1715," ISBN: 0415160731, 2010.
D. Parrott, Strategy and Tactics in the Thirty Years' War, in "The Military Revolution Debate".
D. Eltis, The Military Revolution in sixteenth-Century Europe, ISBN 0-76070-765-0, 1995.