An 1988 Camaro won't start fuel pump doesn't kick on if ignition is in run but works when tested?

What should happen when you turn the key to run is the fuel pump relay should enable to pressurize the fuel system. And then if you do not start the car immediately it will shut off once it has pressuriezed the system. Once you start the engine the pump should kick back in to supply fuel to run the car. If everything has worked so far and once you start the car and it dies right after it starts running check the oil pressure sender. If that sensor does not detect oil pressure it will not allow the fuel pump to keep running and the engine is supposed to die. This is great if you ramp a railroad crossing and take the sump of the oil pan but not any other time. Get that sensor replaced and all should be well