I have no idea on how getting stuff out of the armory works. Most people I know who own guns either live in housing (not in the gated area of the base) and just have to tell the base police what guns they own. Or if they live in the BEQ rent a locker at the range, they have access to it whenever the range is open.
Also every base I've been on does random car searches. Open hood, trunk, any storage areas, give them insurance and registration.
There have been both command wide and random "health and wellness" inspections of the BEQ. You open your door, they look through anything they want with you standing there, open panels in the ceiling, look under mattress, whatever. I know people who went to NJP for handguns, long knives, flushing agents, ect.