We have two options
Option one: the town name is Vault 104 (or any other number)
backstory - 16 years passed since the vault door opened for the first time, it's residents began to build a small village around the entrance. Not soon after it became one of the largest towns in the area.
The bunker will be used as a vault where merchants can set up stands and the council will rent rooms to wasters
Option two: The Brotherhood of Steel found an abandoned vault and fortified it, not soon after wastelanders began to flood in the area seeking for protection. They began to build around the Vault, a small settlement called "Vault Town"
-Bunker will be Brotherhood only and will be there base, while the outside is public and under the Brotherhood protection.