Do you have a part number for the motor or at least a photo? Is it a 50v or 110v motor? Considering it's a Bally, I imagine it's a motor that was also used in Bally's many bingo games of the same era. The bingo motors show up on eBay regularly. If you only need to replace the stator (the winding part), probably any motor with the identical voltage (50 or 110) would be OK. You could pop the stator off a replacement motor and install it on your Ball Park. That way you wouldn't have to worry about differing gear ratios. But a photo would be a great thing to post, and a part number if you have the user manual and it lists the motor.
Sweet looking game, by the way!