Hey y'all!
I've got a 1991 F150, 351W/5.8L engine. About 125,000 miles. Automatic trans, 4x4.
I can start the truck by using the trick of using a screwdriver to jump across the starter relay, and the truck starts just fine. But I can't start it with the key.
The key does turn on the electronics and fuel pump.
I've already replaced the starter relay, and that didn't help.
I've taken a look at the ignition actuator - those often break. But this one is fine.
I've tried jiggling around the shifter and putting it in gear or in neutral. No go.
One more thing - this started out as a more intermittent problem. The truck would start with the key when cold, but after that it had to sit for a while before it would start again with the key.
So I suspect this is an electrical problem - but where do I go from here?

Grab a test light and clip its alligator clip onto the Negative battery post clamp.
Pull off the Red/Lt Blue wire connector from the starter relay.
Hold the test light's probe on the metal terminal in that wiring connector and have somebody hold the ignition key in the Start position.
The test light will light up if it's getting power like it's supposed to when the key is in the Start position.

If it isn't getting Power, put the connector back on and go under the dash to access the ignition switch.
Bring a straight pin and the test light with you.
Clip the test light's alligator clip onto a good ground point under the dash.
Pierce the wire insulation on the Red/Lt Blue wire going to the ignition switch connector with the straight pin to make contact with the wire strands inside. Hold the test light's probe against the pin as you turn the key to the Start position.
If the ignition switch is good the test light will light up. If the ignition switch is faulty it won't.

If the test light lit up there, then go down to the Backup/Neutral Safety Switch on the transmission.
Clean up a good Ground point and attach the test light's alligator clip to it.
Pierce the White/PNK wire with the straight pin like you did before.
Hold the test light probe to the pin and have somebody hold the ignition key in the Start position.
If power is reaching the Backup/Neutral Safety Switch from the Ignition Switch the test light will light up.
If you have Power there, then pierce the Red/Lt blue wire with the straight pin and have somebody hold the ignition key in the Start position. If the Backup/Neutral Safety Switch (and/or it's wiring connector) is good the test light will light up. If it's faulty, it won't.
((you don't disconnect the wiring connector for this but the picture on the right shows the wires pretty good so I posted it))

Here's a look at how the Backup/Neutral Safety Switch (MLPS) works inside.
Power comes in on the WHT/PNK wire from the ignition switch.
When the transmission is in Park or Neutral the internal switch makes contact and power goes out on the Red/Lt BLU wire to the Starter Relay

When you're done get some liquid electrical tape and dab over any places you pierced with the straight pin to prevent corrosion from getting into the wires
WOW, what a great write-up! Thank you, jack62!
And, it looks like I've got the issue solved, and it turns out it was really simple. The first step - the red/light blue wire attached to the relay. I pulled that off to perform the voltage test and noticed it had a solid layer of corrosion on it. I went ahead and did the voltage test - it was getting power. So I cleaned it a bit and put it back on snug and now the truck is starting right up.
That is fantastic! Thanks for updating us - this could help someone else out in the future.
Probably the 32-year-old ignition switch or its wiring.