Thank you for asking your question on Just Answer.
From your discription this sounds like you may have a bad PCM (computer) the PCM controls the Fuel pump relay thru ground switching.
Access the Powertrain Control Module (PCM). Leaving it connected check for battery voltage on pins 37 and 57 Red wires. If battery voltage is not present, check for a problem at the PCM power relay. If battery voltage is present on the Red wires, check for 5v on pin 26 Brown/White (BR/WH) wire. If the voltage is not 5v +/- 0.1v, disconnect the Throttle Position Sensor (TPS) and Differential Pressure Feedback EGR (DPFE) sensor one at a time to see if one is shorted and pulling the voltage down. If the voltage is low and returns to 5v after disconnecting either sensor, replace the shorted sensor. If the 5v reference voltage (vref) is correct, check the PCM grounds on pins 20 Black (BK) wire, 40 and 60 Black/White (BK/W) wires. If the grounds are good, power feeds are good, and the vref is at 5v, suspect a faulty PCM.