I have been up and down Vault 22(where the quest takes place) and I can't find the scientist, can't find the computer I should be downloading records from, nor can I find out how to open the sealed door to the caverns.

You need to get down to level five, Pest Control. The computer room is easy to miss but it's right next to a barricade so check them all (both sides) for a door and you should be able to find it.

There's another room on that floor with a computer terminal or something that's been pulled away from the wall. Follow the tunnel behind it to find the scientist.

Yes, Keely is in the cave section. You do not have to get through the locked door to find her.

In order to go down to Pest Control I have to fix the elevator first that is something I can't do since it requires a repair skill of at least 50 while my repair skill currently sits at 18. Is there any other way down there without having to use the elevator?

Not sure if the stairs go all the way down to level 5 but I assume you can because when you find Keely, she fixes the elevator if you cannot, but compliments you for doing it if you have already done it.

In one of the rooms in the general quarters there is a keycard for that locked door to the caves on level 2, get that key and go through the caves to access level 5. This is also how you access the room on lvl 2 for parts for a certain BoS quest.

I found it, in the commons area on the bottom floor there is a hallway leading from it with two personnel quarters on both sides with a venus flytrap like enemy at the end of it at the top of the steps. The key card to the cave door isn't in any container but is a world rendered item tucked in the corner of the first room on the left behind some foliage.

I repaired the elevator, but never used it until Keeley did. You can definitely find her and the data without the elevator.