Once you get in the Vault, there's a door to go to the caves on the third floor that you can't open even by lockpicking. You need to get a keycard that's in the living quarter's on the 4th floor (it's red and on a shelf in one of the rooms). Once you get the keycard, open the door to the caves on the third floor, take the path to the right at the first intersection you come to (it should be pretty much right after the door. Then go through the door (I think it technically takes you up to the 2nd floor. Take that path, which will take you into a room with lockers in a couple of the corners. The filters are in one of the lockers. (I think it's back right from where you enter the room.)