I’m not sure what’s been covered in MgT, but from other versions of Traveller the Imperial Shipping Commission (part of the Ministry of Commerce) would seem like a good bet.
The way I see it working IMTU is that blocks of tail numbers are allocated out to individual ISC sector offices. Then, when a shipbuilding project is started, the construction yard applies with an interim registration to its local ISC sector office for a tail number. This should arrive before construction is complete, at which time a full registration package is prepared. (If, for some reason, the number is not required it can be returned to the pool at this point.) Registration is filed with the ISC which then issues a database update via the SPA at all class A/B/C starports.