Access is very limited. The only way to get on is to go with someone who has a DOD badge or is retired military. As long as they drive you onto the base, you're good to go. BTW, you must have a valid ID to get on. If you are driving, you must also have proof of insurance. Do not speed! Do not carry illegal drugs or firearms (even if you have a pistol permit) onto base as they randomly inspect all incoming cars.
If you can't find anyone (Lavonne doesn't have access btw), UAH and Indian Creek are the 2 new "pro" courses, but mowing is spotty. For a fun pitch and putt round, play the oldest course still in its original location at Brahan Springs. There's always a crowd. You might just be able to show up there and ask around to see if anyone can get you on base.