I would take to the Vault, for sure. But, if I were single, I'm not really sure I would. Stuck in a hole with a bunch of strangers I will be forced to try and like, or at least, survive with? I'd rather take my chances with the bombs...
Like I said in my OP, there is no catch, Vault Tec simply asks you if you (and by extension your family) would be permitted, it's a standard control vault in the commercials, at some point in time, it will be opened, but it is never specified it will be within your lifetime, would you be willing to live and die in the Vault? Or just try and fight outside?