Once Scribe Rothchild has given you the task to enter Vault 87. You need to go the Great Chamber in Lamplight Caverns, the main computer terminal (no power) and doorway (locked you need a key) into Vault 87 which you find in the Great Chamber is is red herring, it will not open, you need to find a kid called Mayor MacCready, talk to him and he will open the gate to Murder Pass, this is the way to Vault 87 and lots of Super Mutants to kill on the way through the pass so watch out.
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You have to find a civilized super mutant named fawks to go in for you and get the G.E.C.K. He is locked in a cage somewhere in the vault. There is no way to get it yourself because you get in such thick radiation that you die in 1 second. I think max radiation is 400 rads per second.
It's inside a place called Smith Caseys garage. You go there in the main quest.
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if you pluck up the courage and gain about ten levels head west from megaton until you see a small garage guarded with a mr. gutsy robot that is smith caseys garage.
hope ive helped
You must convince the Little Lamplighter's you are not a "mungo" either with a Speech challenge, the Child at Heart perk or rescuing the other Lampligher's in the quest Rescue from Paradise Falls.
All you need to do to get out of Vault 92 is go out the way you came. There is a glitch however where is you close the door then go inside the vault, if the door closes while you are in the vault you are stuck there forever. There is no other way out unless you can glitch your way out. But if the door is still open just go back the way that you came.