Talk to the guard to be able to join Vault City

Go to Vault City and talk to the guard to the right of the inner city gate. Depending on your stats and skills he will require different things of you in order to join. If your charisma is 10, you can beg him to be let in successfully. If this is not the case he may give you a quest to retrieve the Hydroelectric Magnetosphere Regulator from the reactor in Gecko to gain citizenship. If this quest is not available having Repair or Science at 150% will gain you access.

Reward: Joining the Vault City faction and access to the inner city and vault.