Unfortunately, I only know about this question for the Imperial period, so I apologise if you also wished for information about the Republican period.

There were roughly 30 legions active at once during the Imperial period, with roughly 5,000 soldiers in each that's an approximate total of 150,000 active Roman soldiers at any given time. So you are very right in assuming that the Romans were constantly producing new equipment for new recruits. During the Roman period, there were different ways the Romans produced equipment to supply all of their soldiers.

In the early Imperial period, the Romans used the private sector to produce their equipment. This could have been citizens in the city or small town blacksmiths in newly conquered territory. This had been standard practice since the classical Greek period with many wealthy citizens owning weapon 'factories'. By the second century CE, the production of arms had moved 'in house' with the introduction of fabricae into the Roman camp to produce the equipment needed for each legion. Then finally by the fourth century CE these fabricae moved out of the camps and became centralised, state-run, weapons factories that would be able to supply the entire army. It must be kept in mind that in all of these practices slaves were used, meaning that they could produce equipment quickly and cheaply.

However, the Romans 'cheated' with how they supplied new equipment to new soldiers.

The equipment used by Roman soldiers was owned by the soldiers, however, the majority of soldiers could not afford the large costs that came with buying a full panoply and other supplies required to be part of the Roman army. So, the army would make an advanced payment needed for weapons and equipment to the soldiers, which the soldiers would pay back with deductions from their salary. The soldiers could have used this advanced payment to buy their equipment from local smiths, but the advanced pay was not enough for a full new panoply. Fortunately for the new recruits, the army had a backlog of preowned equipment which the army bought back from soldiers either upon their retirement or death. This allowed new recruits with little money to purchase their equipment for a lower sum of money.

We know that this was a standard practice due to the large amount of equipment we find with inscriptions of ownership. A lot of these finds have multiple inscriptions, for example this helmet in the British Museum had at least 4 different owners.

This meant that the Romans did not need a high rate of production as there was only a small trickle of soldiers joining and leaving the army at any given time and only created new equipment when their old equipment was badly damaged or lost.

Sources:

M.C. Bishop, 1985. 'The Military Facrica and the Production of Arms in the Early Imperial Period'. In M. C. Bishop (ed.) The Production and Distribution of Roman Military Equipment: Proceedings of the Second Roman Military Equipment Research Seminar. BAR S275. Oxford

R. MacMullen, 1960. Inscriptions on Armor and the Supply of Arms in the Roman Empire. American Journal of Archaeology, Vol. 64, No.1, pages 23-40.

J. Nicolay, 2007. 'Military Equipment and the Life Cycle of a Roman Soldier'. In: Armed Batavians. Amsterdam University Press.