The door to the room is marked "Locked - Requires key" but i can find no key and the wiki makes no mention of any special key needed...

From just before the locked door, follow a hall to your left and you'll find a room with a computer that grants access.

If I remember right the Vault passage leading to this comes to a T junction, if you go right you have the locked room with the prize inside, if you go left you will be able to access the computer in a different room and unlock the door (you can see the room & door you are unlocking becasue of windows between the two rooms). I can't remember what they called this level of the Vault though.

Living Quarters. Use the sound flush to insta-kill all enemies on that level.