Assuming the bulb lights up when the engine isn't running, engine vibration is causing a break in the circuit somewhere between the fuse (assuming you have 12v at the fuse) and the bulb filament (assuming you don't have 12v at the bulb).
1) check the bulb filament or replace the bulb with a known good bulb, note its a dual filament bulb.
2) check the fuse for continuity or replace the fuse with a known good fuse.
3) check for loose connections, has the loom been extended via a relay triggered from the rear light?
4) check the circuit, multimeter in hand check for 12v from the bulb holder and work back through each connection to the fuse.
5) check for ground to chassis.
Note all checks will need to be carried out with the engine running. If I was a gambling man I'd wager a failed bulb filament.