There's an Alternator relay probably near the Ball Count unit that should fire on ball 2 and ball 4:
Grand Prix Alternator relay (resized).jpg
I suspect either your Alternator relay isn't firing at all, or it's firing as it should but one of its Make/Break switches isn't toggling properly.

If it's not firing at all, trace the path through the Ball Count unit looking closely at the wipers. There's a drawing of it on page 14 in the manual on ipdb.org.

If it is firing during ball 2 and ball 4, try cleaning the contacts of the Make/Break switch(es) on the Alternator relay.

/Mark

Check my thread here post #27 and on. I'll follow along if you can't get it going again.
https://pinside.com/pinball/forum/topic/william-s-grand-prix-ball-wont-kick-out-end-of-ball-wont-advance-

In your case, it might be that same switch on the right bonus relay. Use a DMM to check continuity between the wire lugs on the blades of the switch contacts.

There are a couple of ways. But before making a bunch of adjustments be sure that the stepper is clean and operating well. It should snap cleanly forward and backward when you push in the solenoid plungers and let them go.

Your last photo barely shows the two mounting screws that hold the contact board to the frame. They're under the wiper arms. If you remove the wiper arms you can loosen those two screws and rotate the contact board a little bit. That may be all it needs.

If that doesn't work, the other side of the frame has an adjustment screw to regulate how far the stepper arm pushes the wiper disk:
Williams stepper adjustment (resized).jpg
Loosen the lock nut and move to bolt in/out to more the gear less/more. Once it's dialed in so the wiper arms come to rest in the center of the contacts tighten the lock nut again. I usually find that cleaning and/or contact board rotation takes care of it. Those lock nuts don't come loose often so that's less likely to be the issue.

The smaller solenoid on the right side is the reset solenoid that lets the gear and stepper arms step back or reset. The larger solenoid on the bottom is the advance solenoid that moves the gear and wiper arms forward. Apart from doing everything the reset solenoid does as far as moving the gear and wiper arms is concerned, the advance solenoid must also wind up the torsion spring that is wrapped around the gear shaft. That torsion spring is what lets the gear and arms step back when the reset solenoid moves the arms out of the way.

If you push in the advance plunger and the stepper gets stuck you can help it along to complete its stroke. If the wipers end up over the contacts after you've nudged it along it's unlikely that your stepper needs adjustment.

Your stepper is likely binding because of added friction due to dirt, grime, old lube and/or wear. A good cleaning may be in order. It might also be a mushroomed plunger or worn coil sleeve. I'm pretty sure there are other topics that cover stepper cleaning.