I have never worked under the hood of a Travel Time, but have played one before.

My "guess" would be that Williams is calling that the shooter coil and it is likely getting a weak pulse due to dirty switches in the circuit. You should be able to confirm by checking the wrapper on that coil (if it still has one) and seeing if it is a G-22-550. If so, you'll want to concentrate on the circuit shown at column 9 on the schematic, starting with the level C switch on the #5 cam and follow with the switches on the shooter relay and the special relay.

Interesting that your photo shows the version made by Segasa of Spain. Is that the machine the gentleman owns?

The switches at Level B of the Index cam pulse the shooter relay, eject relay, and a couple other relays.
It is possible the shooter relay is not getting a strong enough pulse to firmly close the NO switch that passes current to the shooter coil, but I would concentrate on the circuit in column 9 first. Also check the leads to the shooter coil itself as a fractured wire or solder joint could be resulting in a very weak signal to the coil.

I think the schematic text for the score motor switches is a little misleading.
The way I interpret the schematic is that the Index cam level B switch pulses (among other relays) the shooter relay, not the shooter solenoid.

If it were me, I would first inspect the shooter solenoid coil itself and make sure it is the correct number and then check the tabs where the wires attach to make sure neither wire is loose, broken off, or has a fractured solder joint.
Next, clean and adjust the switch at level C of the #5 cam, and finally, clean and adjust the switches in the shooter relay and the special relay.

You could use Howard's technique and jumper around the latter 2 switches to see if either makes the shooter coil kick normally.