How do I enter Vault 87?

I got stuck there too, find the mayor kid and talk to him and repeat the conversation, there should be an option now letting him open it for you.

After mayor try to talk with kid in Great Chamber. Don't remenber exact name, maybe Johnny. Wear white T-shirt. Speak about forgotten password to the computer. It is much easier, than go thru mayor's path. Req 50+ science skill fo hack terminal.

either talk to the Mayor several times, and convince him to let you through the back entrance which will lead to Murder Pass, this route has about 10 super mutants inside. Alternately find Joseph in the Great Chamber and ask him to turn the computer back on for you, hack the password and you can immediately go into the vault''s Reactor level.

after i got inside the tunnels i went to the back were the gate is and talked to the guard about the gate etc. then i went back and talked to the Mayor about opening the gate.

Apparently there is also a garage located SSE from the Vault, you can find a switch which opens the lower level, allowing you to access the vault.

You can also speak to the Mayor again - there's TWO ways into the vault. That door is one, "murder pass" is another.

ok first you go to Lamplight and talk to the kid at the gate he'll tell you to find his friends so then you need to go to Paradise falls and find and capture two slaves that the guard requests of you after you get the slaves you can enter then you'll have to find a way to break out the Lamplight children from the cage after you get them out go to Lamplight and they'll let you in. You will then have to go through Murder Pass which the kid at front gate will show you where it is in Lamplight and open the door for you... Once inside its kind of like a one way path but when i did this i went in to the test lab and got kidnapped by the enclave and taken to Raven Rock(enclave base top most left of map) but if you go back you may beable to continue on through the place and eventually reach Vault 87

ok first you go to Lamplight and talk to the kid at the gate he'll tell you to find his friends so then you need to go to Paradise falls and find and capture two slaves that the guard requests of you after you get the slaves you can enter then you'll have to find a way to break out the Lamplight children from the cage after you get them out go to Lamplight and they'll let you in. You will then have to go through Murder Pass which the kid at front gate will show you where it is in Lamplight and open the door for you... Once inside its kind of like a one way path but when i did this i went in to the test lab and got kidnapped by the enclave and taken to Raven Rock(enclave base top most left of map) but if you go back you may beable to continue on through the place and eventually reach Vault 87