I'm army and my girl and i stayed on base in the hotel and shopped at the exchange and grocery store. Just don't go to the mess hall. I have my tan card.
As long as you have done form of ID that allows you to get on base, you can go. No matter what branch.
Once you get the ID card, ask or check to see what you are entitled to. But generally, you can go to do shopping. Just ask where you did your paperwork or go to where you got your ID
Thanks! Your answer is awaiting moderation.