If you are over level 20, Vault-Tec Calling will start when your Pip-Boy detects a distress signal from Vault 88. The vault is in the southeast and is accessed through a small cave beneath Quincy Quarries.
Clear the area of raiders and make your way down the stone steps to the lower bowels of the quarry. Fight through some more raiders. Enter the vault, and lay waste to a few more raiders who are attempting to break in the vault. Activate the vault door controls which will complete Vault-Tec Calling and trigger Better Living Underground.