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.
I stumbled upon a possible solution, written by user anonymous4761 on the GameSpot forums (and slightly edited), which involves using Power Armor to force yourself into the restricted space of the Vault:
Put on Power Armor you usually never use (strip it of all useful parts, too.) There are windows in-between the entrance area and the security room in which a security guard sits. There is a flat metal platform above these windows. Use the mechanism to the right of the inaccessible door to jump onto this platform.
Here's the trickiest part: your goal is to step out of the Power Armor and fall backwards into the security area. Because you can't face away from the wall and step out of the armor, you have to start facing forwards, hold the "Exit Armor" button (PS4: ✕ / PC: E / Xbox One: A), and spin around quickly. It might take several tries, but I've found that standing on that metal piece above the window makes it work better.
Once you make it inside the actual Vault, you still won't be able to open the door - even completing the Hole in the Wall quest (which opens the terminal-locked door) won't trigger the inaccessible door to open. So you have to use a trash can or ceramic bucket inside the security area. Pick it up, go to the corner by the inaccessible door, and jump and move backwards into the corner.
This is a pretty commonly used wall-bypassing glitch, and it shouldn't take long to get back into the vault entrance area.
There are two ways to get back inside the vault. The first way is to complete the Here Kitty Kitty quest, which will cause the cat to run back into the vault. The door might open when he runs through.
However, in my game the Here Kitty Kitty quest yet again has a bug which prevents you from interacting with the cat and sending him home. So you'll have to use your Power Armor. Jump up onto the metal platform, but don't step into your Power Armor. Instead, hit "Transfer" and withdraw the power core, then put the core back into the armor. When you do that, the animation will cause you to end up behind the Power Armor, and you can then move backwards and fall into the vault (this is also why you should use junk Power Armor - it'll remain the only way to get in and out of Vault 81).