I doubt it - and almost certainly not worth the hassle. I do have authorized access to the Military Uniform Store only (with Civil Air Patrol / US Air Force Auxiliary ID). A few years ago the prices in the Nellis AFB BX were not significantly cheaper for anything than in large chain shops in Las Vegas. It's nothing like the savings you could get at Ramstein compared to civilian shops in Germany or the UK. It is more there as an on-base convenience.
Feel free to PM me if you want a good steer on prices / store chain name for anything in particular such as clothing, cameras etc. I've pretty much started from scratch after immigrating here 7 years ago.