1991 F150 5.0L, The fuel pump relay clicking

Year make and model ?
Have you swapped the fuel pump relay with another ?
Its normal for the traction control light to come on if the check engine light is also on.
Have you checked the engine computer for trouble codes ?

How do you know the fuel pump is good ? Did you do a fuel pump pressure test with a test gauge ? Did you test the fuel pump electric circuit ? Is the check engine light on ? There is a fuel pump relay under the hood engine compartment . The right front behind the headlamp are four relays , the one to the left is the fuel pump relay .I'll send a link to a video on testing the circuit !There also is a fuse in the wiring near those relays , it's power for the fuel pump relay ! Or you have a bad fuel pressure regulator ! A plugged catalytic converter could do this , need to pull the O2 sensor an do a back pressure Test ! Fuel Pressure Test Exhaust Backpressure Testing How To Diagnose Exhaust Restrictions EricTheCarGuy

Relay or fuse?

If you have no fuel pressure at the engine when key is on, the pump is not coming on. Check fuel pump fuse, check fuel pump relay (relay should click when key turned to on-put finger lightly on it and feel for click when someone turns key to on). If no click, signal from computer may be lost, ground for relay may be lost, or relay may be bad. If it clicks, wiring from relay to pump may have failed, or the relay's main power source may not be there at relay-find the fuse link or maxi-fuse that powers that relay. One terminal at relay should be hot at all times-can check for that with a test light.
If pump is working and you do have fuel pressure, you will need to check injector circuit. Post back if questions, or need further help.

The VATS system is not supposed to let the engine crank when it sees the wrong key code.
I would check the pump relay and see if it has power and ground after the engine starts.
The computer has to see oil pressure and/or RPM to power the pump relay after the engine starts.

This is a delay to power the fuel pump. You need to check the output from the ECU/ECM. check for any error codes, check if there is power from the switch. Once the engine runs , you need to see if there is any error codes or faulty symptoms.

Had a problem like this before and it was the computer..But you could faintly smell something like burnt wires..The computer should be down by the kick panel on the driverside..See if you smell anything on that side

if it starts using ether, it suggests a clogged fuel filter. You need to know what the true psi value is across the fuel rail. you can rent kit from major auto parts stores. be it remembered that many cars nowfays have 2 fuel filters- one in tank and one on frame or firewall. a quick check is back pump air through your fuel line till you hear bubbling. recheck fuel pressure and note diffetence. if it goes up you probably icked up larger debris on the screen.

check your relay and your fuel filter should be replaced every 30,000 miles

fuel pump relay is in engine compartment on drivers side under the flap the two relays that are together ,it is the bottom relay,,u cant by pass.if relay is good ,if it clicks when turning over and key on and off (it should click and go off if key is turned on) then use a test light and ground it find wires going to top of tank and probe wire have some one turn key on ,it should light up test light and go off ,if this happens ,replace fuel pump