If they are building that many in the sector per year I'd have the IN set up production at dedicated Depot/ship yards that are NOT civilian. They can forecast production and bring in needed personnel on contract. It would not be that bright to try and get shipyards at civvie starports to carry that load. If for no other reason than security and secrecy.
I agree that the builders making Navy warships are most likely specialized military builders. They have to deal with armor and specialized weapons and defenses that a civilian builder would never need to work with. I presume a civvie shipyard would be more than happy to make an unarmored transport or tanker if it had capacity laying idle, and I don't see any reason the Imperium couldn't farm out contracts for its little workboats to B ports in order to stimulate those economies, but I presume the warships are built by builders skilled in warship construction.
We know where the depots are. There are 19 serving the major sectors. Some sectors, like Spinward Marches and Verge, do not have a depot. Delphi gets two, which is odd considering the size of the sector and the lack of an obvious enemy of significance. Depots are described in MT as being for maintenance "and even construction," which to me implies that they do it but that it's not a primary role. I would have expected them to play up the construction role more if they were at the center of the fleet's shipbuilding program.
AHL tells us it took only about 2 1/2 years to complete a Lightning class cruiser, which highlights that I forgot about the 40% reduction in time for building ships of a class. At 25 ships per year per sector average, that changes the figure to about 60-ish ships under construction at any one time in any given sector.
We know of 21 builders from AHL. Some, like Clan Severn and Yard 17, were also mentioned as building the little Kinunir. However, there are two yards mentioned in Kinunir that did not participate in production of the AHLs: Mars and General. It's possible those companies didn't exist when the AHLs were being produced, or their yards weren't able to handle ships in that size range.
We do not know where those builders did their work, except that the Duke of Regina worked a deal to get some of the production done in the Marches, which does not have a Depot; those were built by Ling Standard. Most of the builders made multiple ships, with the start of the next ship not occurring until after the completion of the previous one, usually within days, implying their full resources were involved in the construction of the ship. Three builders either were not given the opportunity to build more than one or decided to forego the opportunity.
Some of these builders seem to represent one corporation with several sites: Gashidda 1, 2, and 3, for example. Those sites may be in different systems or all in the same system, we don't know; there may be legal or organizational advantages in having a corporation use a distinct subsidiary for each slipway. There may be more than one corporation in a given system; again, we don't know. It's clear, given that these 21 builders supposedly represent builders across the Imperium and that there's something like 1200 ships under construction at any one time across the Imperium, that these represent only a small sample of the builders doing work out there.
It is possible that all these builders are doing work at the Depots, but it strikes me as more likely that they're working around worlds that have the technological and industrial base to provide them with the tools and construction equipment they need for the work. A lot of these Depot worlds - most of them - seem to lack that infrastructure; everything is Imperial Navy and in deep space (where they don't appear to get counted toward the local population), and their primary worlds don't seem to be much more than R&R spots. My guess then is that the major Navy shipbuilders are located at the major A ports, places like Trin and Mora, Rhylanor and Glisten, perhaps Palique, Efate and Lunion if the Imperium gets auxiliaries built at lower tech.
An interesting point is Kinunir, constructed at Regina (now TL12) using Book-2 construction and then revised to High Guard specs, is rated at TL15 and includes features such as a TL13 jump drive (by errata), a TL15 power plant (same), a TL14 nuclear damper, a TL13 computer, TL15 particle beam turrets, black globe generators (TL15), and by the errata TL13 missile systems and x-ray laser turrets. The ship is unarmored, so no saying whether that's an issue or not, but it's pretty clear from the example that importing major equipment for ship construction is not an issue with at least the escort-size ships.