When I get past the main door at Vault 81 and talk to the Overseer, I can't go any further. The door into where the elevator is is inaccessible. What am I doing wrong? Do I need to complete a quest first?
It looks like this is what is happening, since you are playing on PS4 and passed the speech check to enter. From the wikia page for the Vault 81 quest:
When you pass the persuasion check there is a chance that the door used to head deeper into the vault will not open and remain inaccessible. There is no currently found fix for the issue.
As mentioned in the quote, there is no fix for this as of right now.