Why would they want to come out? They are no longer rich. Apparently Vault Tec can build vaults that can last for many decades.
Because you are alive. Sitting in a vault the rest of your life, would be like a voluntary prison sentence.
So, it depends on who the "vault" is for. If for a corporation, and they have scientists, they would maybe come out to rebuild. A politician, may want to come out and rebuild and rule. Just because they emerged from their shelter, doesn't mean they can't keep using it, like Vault City.
Anyway, can be a lot of reasons a group would emerge, depending on who was there, their goals, etc.