Fuel pump wont start when ignition is on unless you hold fuel relay switch

Have you checked that your inertia switch isn't tripped?

This should be in the owner's manual. I found the below on autozone.com, their repair help location:
An inertia switch is used as a safety device in the fuel system. The inertia switch is located in the cab, generally under the dashboard on the right side. It is designed to open the fuel pump power circuit in the event of a collision. The switch is reset by pushing each of 2 buttons on the switch simultaneously (some models use switches with only one reset button). The inertia switch should not be reset until the fuel system has been inspected for damage or leaks.
When the ignition switch is ON , it turns the EEC power relay ON . The EEC power relay provides power to the powertrain control module (PCM) and the control side of the fuel pump relay. Power for the fuel pump(s) is supplied through a fuse link or high current fuse attached to the starter solenoid (battery side). From the fuse link or high current fuse, current flows through the fuel pump relay and inertia switch to the fuel pump(s). The fuel pump relay is controlled by the PCM.
When the ignition switch is turned ON , the fuel pump(s) will operate. If the ignition switch is not turned to the START position the PCM will shut the fuel pump(s) OFF after 1 second. The PCM will operate the fuel pump(s) operate the fuel pump(s) when the ignition switch is turn to START position to provide fuel while cranking.
After the engine starts, the PCM will continue to operate the fuel pump(s) unless the engine stops, drops below 120 rpm or the inertia switch is tripped.

Fuel pump relay, turning the ignition on runs the fuel pump for about two seconds to pressure the fuel rail to start, then the fuel pump relay takes over once engine is started, it relies on a signal from the ignition or ECU (to say keep running). Either the relay has gone down or the relay has lost the signal. Don't bypass the relay, it's a safety feature, to stop the fuel pump if a fuel line ruptures.

Sounds more like ignition switch to me.

When you say the fuel pump relay is good did you test for power to the relay circuit ?

Check for fuel pressure on the fuel rail while turning the key to start,if there is fuel pressure(enough)then the injectors are not pulsing(allowing fuel into the intake).The pick up coil in the distributor,will be the problem,unless it is distribuless ignition,then the cam sensor would be at fault.

Turn the ignition key to the ON position without starting the engine

You should hear the fuel pump priming for about a few seconds then shutting itself off.
If not hear replace the engine oil pressure switch
The 3rd wire to that switch is the pumps prime switch lead.
When the engines oil pressure is reached the computer will shut that circuit off and turn on the fuel pump relay
If you hear the pump prime the relay maybe at fault instead

There is a fuse for the fuel on the firewall inside the cab:




Fuel Pump The fuel delivery system uses either a high or low-pressure in-line or in-tank electric fuel pump, with some models equipped with both. It is a recirculating system that delivers fuel to a pressure regulating valve in the throttle body and returns excess fuel from the throttle body regulator back to the fuel tank. The electrical system uses two types of control relays, one controlled by a vacuum switch and the other controlled by the powertrain control module (PCM) to provide power to the fuel pump under various operating conditions.

An inertia switch is used as a safety device in the fuel system. The inertia switch is located in the cab, generally under the dashboard on the right side. It is designed to open the fuel pump power circuit in the event of a collision. The switch is reset by pushing each of 2 buttons on the switch simultaneously (some models use switches with only one reset button). The inertia switch should not be reset until the fuel system has been inspected for damage or leaks.
When the ignition switch is ON, it turns the EEC power relay ON. The EEC power relay provides power to the powertrain control module (PCM) and the control side of the fuel pump relay. Power for the fuel pump(s) is supplied through a fuse link or high current fuse attached to the starter solenoid (battery side). From the fuse link or high current fuse, current flows through the fuel pump relay and inertia switch to the fuel pump(s). The fuel pump relay is controlled by the PCM.
When the ignition switch is turned ON, the fuel pump(s) will operate. If the ignition switch is not turned to the START position the PCM will shut the fuel pump(s) OFF after 1 second. The PCM will operate the fuel pump(s) operate the fuel pump(s) when the ignition switch is turn to START position to provide fuel while cranking.
After the engine starts, the PCM will continue to operate the fuel pump(s) unless the engine stops, drops below 120 rpm or the inertia switch is tripped.

Try your inertia switch in the cargo area in back. Its located on the passenger side rear wheel well. Push the button.

no fuel issue. check fuse for fuel pump check fuel pump fuel pump relay fuel filter could be pluged.if you turn engine over acouple turns and stop you should hear fuel pump buzz.also check wires at ignition coil for corrostion or broken fuel pump relay gets its singnal for negitive side of coil.