Registering with the VA does not necessarily get you use of military recreational facilities. If you are eligible, you will likely be carrying a DD Form 2 ID Card, rather than a document from the VA.
Some recreational facilities are limited to the same groups eligible for commissary, exchange and medical care privileges (that would be military hospital and TriCare, not VA). Others are open to a larger audience, some even to the public at large. This is at the facility commander's discretion. Rules are often different for different recreational facilities, the golf course for example may be more open than a family campground, while the auto shop could be more limited. It somewhat depends on funding and fee structures, there are rules attached to some sources of funding.
Not all of the recreational facilities are on base. Campgrounds and marinas, particularly, might be on a separate property some distance from any base, like the Lake Texoma FamCamp serving military personnel at bases in northern Texas and southern Oklahoma.