I think they have some keyed patio door locks that you could put on the 2 end doors. Maybe put one inside & the other on the outside. You might have to channel your inner Macgyver
A locked closet door is a dead give away that there is something valuable inside. You'd be better off bolting a small Stackon cabinet to the floor or wall for the ammo and putting something in front of it so if a baddie peeks in the closet all he'll see is your prom dress and ed hardy t-shirt collection.