Quoted from Egress:
this delay relay uses a capacitor, not a bulb, and doesnt seem to be a part of the coin door circuit, if i trigger the relay with a pencil the score motor engages and acts as though the firecracker unit has just stops spinning.
is the capacitor only supposed to delay a small amount?
As mentioned above, the delay is meant to stop an angry "person" from continuing to abuse the machine by kicking, lifting & dropping or slamming it against the wall. The delay should be 5 or more seconds.
Look closely at the schematic diagram and study the circuit of the Delay Relay. Be sure your game is wired exactly as shown. The bulb socket is supposed to be attached to the relay. No bulb in socket means no delay. Remember, a #455 flasher bulb need to be used.
UDATE: Since I am not near my schematic files, I see Rolf provided one below that shows Firecracker uses a capacitor for the delay circuit instead of a flasher lamp. Ironically, Bally's "Introduction to Bally Flipper Games" EM service booklet only mentions the 455 lamp for the Delay Relay circuitry.
It appears that the capacitor may be dried out and has failed although standard electrical troubleshooting techniques should be used to make the repair. On the other hand, there is no need for this circuit to operate in a home use environment nor even in a supervised Arcade that is frequented by pinball Afficionados.