To start with, although population measures are a little imprecise, Eboracum in the third century probably had a population of somewhere between 10-14,000 people, with a minimum fortress population of about 3000 and a maximum of about 7500. No small amount of the remaining population would be retired soldiers who had settled in the colonia, as well as craftspeople, sex workers, soldier’s families, merchants, and so on and so forth. The fort would also have included specialists like engineers, smiths, farriers, arrow makers, plumbers, grooms, tanners, leather workers, shield makers, clerical staff, and so on and so forth - everything you needed to maintain the large number of soldiers and their equipment.
By the third century Eboracum would have been the political centre of the northern “frontier” - hence the death of Septimius Severus and the ascension of his two sons (shortly to be one - dynastic politics is a nasty business) there. It was an important place. The choice to rebuild in stone would have been an expensive one, but also one that made it clear that the fortress was intended to be permanent.
The rebuilding probably started sometime in the early second century, but continued into the third. It was likely primarily done by soldiers, who would have had varying levels of skill, under the direction of military engineers. Although we cannot definitively say that no local or enslaved labour was employed, it seems unlikely. Having an army - who you are already paying - sitting around doing nothing is extremely expensive, and we know that they worked on other engineering projects like road building when not actively pointed at an enemy. Looking at sites like Hadrian’s Wall there is direct evidence that they were responsible for building work in the form of centurial stones - essentially inscribed boasts that “x legion under y commander built this”.
As an additional detail - the building material used seems to have largely been from relatively local limestone quarries. Although the precise quarries are unclear, it’s possible that it came from the Derwent gorge; it could have come down the Derwent and either been taken off of a barge at Stamford Bridge and shipped by road, or taken slightly further down so that it could be taken back upstream on the Ouse while remaining on a barge. We know that the lowest part of the Derwent is an artificial course cut during the Roman period which would likely have cut approx. 14km off of such a journey, as well as avoiding some tricky currents. Other theories have included places like Tadcaster, which is directly on the Wharfe (also feeding into the Ouse) and was definitely a medieval source of limestone.