Vault 13 is one of the major fallout vaults developed by Vault-Tec. It is the Vault Dweller's home from Fallout 1. It is probably located right under Mount Whitney, as it matches quite exactly the mount's location.
The vault's construction started in August 2063 and ended in march 2069. Meant to stay closed for ten years, the vault was also containing 1000 inhabitants, protecting them from radiation of Atomic bombs from the Great War. However, as a study about prolonged isolation, the Vault remained sealed for 200 years, which would eventually lead to problems with its Water Chip.
You cannot go inside this particular Vault in FOnline: Forever. Even if you found the location somehow, you would be restricted to the outside cave because the giant Vault 13 door is sealed. The Vault's door remains closed after the Vault Dweller was expelled.
Vault 13's location would be Grid 34:21 (directly west of Shady Sands, if it were accessible at all).