Reading on The Vault wiki, the Cog Door for Vault 21 ended up as the hotel's sign (though why they opted to use a massive multi-ton nuke-proof hatch as a sign boggles the mind... or even how they got it out of there to begin with).

But what about the cog door hatch and its room? Where'd it go?

Vault 21 doesn't seem to have a normal GIANT VAULT DOOR entrance like the other Vaults. It appears to be just that stairway in the Gift shop.

It could be lower underground where House filled the vault with cement. There could be another entrance to the vault elsewhere where the main entrance was.

Most likely they dismantled the entire mechnisim for the door then hoisted the thing up to act as a sign using a crane. The crane was then dismantled and put into storage for later construction.