In Dead Money, how do I trap Father Elijah in the vault?

It's tricky to do, and he does tend to find you if you get too far away from the vault before he arrives, but it's doable:

[Elijah] becomes instantly aware if the Courier steps near either of the two forcefield barriers nearest the elevator, even if the Courier has 100 Sneak and is using a Stealth Boy.

In order to sneak past Elijah and seal him in the vault, resulting in the Safety Deposit Box achievement, simply leave the vault when Elijah comes down and remain Hidden outside the blue forcefield barriers. When Elijah triggers the trap that seals him in the vault, he will realize you've tricked him and activate your collar.

I found it easier than it sound.

As soon as you persuade him to come down in the vault, start running your way back to the elevator. You have to follow your steps back, since the direct way is closed by blue barriers.

In the meanwhile Elijah will walk to the vault door. He will find out that you are not there, and start talking. You have all the time to reach the elevator and find out that is closed.

You must wait until Elijah finish his talk and your collar start beeping to take the elevator. As soon as you take the elevator the achievement should unlock.

It is very important to follow the proper dialog tree with him otherwise he will attack you and won't go anywhere past vault's door.

Choose any of the following paths, but only one:

• Unless you come down here, I'll destroy the vault, everything in it

• I'm not interested in the vault, I only want to leave.

• Then I'll find a way to unlock the secrets of this vault, take them for myself.

• If you want me to wait, I'll cooperate. The Sierra Madre doesn't matter to me.

After he says he will come down, just go out the vault to the left and crouch. Although it will say you are Detected, he will pass and continue towards vault's door. Now just make a run for the exit.