The Raider brand BB gun is a small lever-action air rifle. The only thing this toy is practical for is killing radroaches.
As it is an airgun designed for children, the BB gun sports the lowest damage of any firearm in the game. This also makes it not very durable and vulnerable to breaking very quickly. However, it has the notable quality of being silent. A fully repaired BB gun can fire about 245 BBs, or about 2.5 magazines, breaking.
Vault 19 - when you enter the living quarters, there's a room on you're left in front of you. In the room is a big pile. If you walk over the pile you see a footlocker containing a BB gun with ammunition.
The BB gun may be considered a nearly useless weapon, because very few enemies are weak enough to be killed by a BB gun, aside from powder gangers and bloatflies.