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82 with a crossfire? Fuel pump relay is energized by the ECM for 10 seconds after key on. Engine starts, ECM sees the pulses from the distributor. Oil pressure comes up and keeps the fuel pump relay on, feeding power to the fuel pump. Fuel pump is in the tank. Guaranteed to not fail unless you just filled the gas tank. Oil pressure switch is just above the oil filter on the left side.
1982 corvette . No fuel to fuel pump after starting.
You check the sender with an ohms meter. You should get a reading of 1-90 ohms. If when you start the car you don't have any ohms coming from the sender then your fuel pump relay will not stay energized.
Depending on what the oil pressure gauge does when you start the car would determine to me which way to go looking for the issue. If the oil gauge moves and shows pressure when you start the car like Tim said, it should energize the relay. If the oil gauge doesn't move at all, then I'd go to the sender and test it on start up.