It's been a while, so things could have changed (hard to keep up with the revolving door of command) but the last known (to me) requirement was that you possess a C&R license. Basically, the military (and JAG specifically) got out of the firearms approval business. If it's OK on a C&R, it's OK.
That said, after several years over here, I have yet to find much that I would want to purchase, even as a souvenir and far less that is reasonably priced. I guess non-gun guys don't realize that you can purchase viable, functional, operating, modern firearms (back home) for what they are willing to pay for broken Arabicized Martinis and worse.
I would ask the P.O. as they are the ones that see them out the door so to speak. Up in Bagram they used to set up at the bazaar with a one stop JAG/USPS shop. I have not seen that anywhere else.
Good luck.