How to open the maintenance door in the sierra madre vault
To leave the Sierra Madre vault you need to backtrack through the whole way you entered the casino. You cannot take the same route Father Elijah takes into the vault room.
This is done under severe enemy pressure, pressure from collar detonation, and movement limitations. This not only makes it dramatic, but also means your courier will be unable to flee the vault with all gold bars (unless using glitches); you can only leave with what you can carry within your capacity.
Instead of going to Elijah's maintenance door, return to winding paths to route yourself out.