You can't have a firearm in your car on a military base. I don't even think you can have one in your home on base. I think you must lock it up in a base armory.
So that would overrule the state probably. I know, this is why I dan't carry one in my vehicle. Nowadays, the last thing I want, is to have my car searched and have a gun in the glovebox found when I come to work.
Do you have any references?
I think you are mistaken.
Plenty of bases offer recreational shooting ranges, trap and skeet clubs, etc. to their military and civilian members. These are typically not within walking distance of base housing, thus it can be assumed a vehicle is needed to transport the shooter and his or her firearm(s) to the range. A car is likely, or perhaps a bus for those without automobiles.
A shotgun shuttle from the armory to the skeet range? Unlikely.
I'm not aware of any requirement that privately-owned firearms be stored in a "base armory" when not in use. In all likelihood, military regulations would prohibit keeping private weapons in a place designated for secure storage of -- let's face it -- extremely lethal weapons whose possession by ordinary citizens would likely be viewed with extreme prejudice.
Perhaps one of our esteemed military FReepers can shed some light on whether the gun-control fanatics have infiltrated the military or not. My hunch is that they have not.