So in a couple of the Fallout Intro's the narrator mentions "humanity survived in underground shelters known as vaults" so are a majority of Post-War people from vaults? I know in Fallout 1 and 2 the major hubs are from vault descendants. But what about raiders? Like the Big Khans etc. The people in The Pitt. I think in FO3 one of the residents in Megaton mentions the city was founded by a vault group. Rivet City was started by people from some Research company.

Are the majority of post-war people from vaults?

No, if anything, vault people make up a minority, since most vaults were designed to fail, and ended up killing everyone in them.

Out of the 122? vaults, only 12 of them were designed to work as advertised, and even some of them didn't work out.

Megaton was founded by the people who tried and failed to get into 101 when the bombs were dropping.

To answer your question, no not everyone is from the Vaults. I don't even think you can say for sure the majority of them are from Vaults. They seem to be a minority, considering how few of the Vaults were control Vaults rather than experimental ones.

The intros are full of hyperboles and metaphors. They're meant to establish the tone of the game and give a rough background, not provide perfectly reliable historical information. As such, take it with a grain of salt.