The schematic diagram would offer some clues. What game is this for?

/Mark

The schematic shows that the Balls Played Unit has 20 positions (0-19). It also shows the connections that are made in each of those positions. The numbered circles represent the contacts in the various positions of the unit and the double ended arrow between pairs of contacts represents the position of the wipers on the wiper board. The schematic is drawn showing the double ended arrow in position 0 or the reset position. As the Balls Played unit steps forward you have to imagine the double ended arrow moving forward to other positions in the schematic too.

To figure out where your wiper board should be mounted we can take clues from the schematic:
Ship Mates Ball Count (resized).jpg
In the top half of this schematic the red highlight shows how the circuit is completed when the Balls Played Unit is in the zero or reset position. In this position there is an electrical connection (little or no resistance when measured with a meter) between the blue-black wire on the left and the yellow wire on the right. The same connection will be made again in positions 4, 8, 12 and 16. So if you reset your Balls Played Unit you should be able to adjust your wiper board until you observe a connection between the blue-black and yellow wires. And if you step the Balls Played Unit forward the connection should reappear every 4 steps.

The bottom half of the schematic shows a circuit that is only complete when the Balls Played Unit is in the 19th or last position. In that position the green-black wire should be connected to the yellow wire, and also to the maroon-green wire on the 3-5 ball adjustment plug (assuming the adjustment is in the 3 ball position).

You should be able to find the wires soldered to the edge of the Balls Played Unit or on the other devices shown on the schematic. If you have a meter you can measure the resistance between the various wires to determine what location the wiper disk should be in for a given position. Once the wiper board is in the correct position you should be able to verify the connections shown on the schematic for all 20 positions.

You might want to clean all the grease and dirt off the wiper disk and contact board before starting though. The step unit should advance and reset crisply and cleanly without any sluggishness before you start your experiments.

The attached photos may also offer a clue also. The image on the left taken from the IPDB and your original image in post.
By lining the linkage with the hole in the disc and contact shoes, you should be able to find the correct position. Unsure whether the IPDB base image is in the reset position. Info supplied by MarkG will no doubt confirm this.

ship mates (resized).jpg

The switch with the orange wire is likely a limit switch to prevent the Balls Played Unit from trying to step past the 19th step. It probably cuts power to the step up coil once the stepper has reached position 19.