Thanks for all the replies and useful info. These units definitely have reversing units- circuit boards with a couple of relays on them.
If those relays resemble a pair of small boxes at one end of the board, that may be a QSIndustries unit. Look for a QSI label on top. To replace it, contact Wayne Renga at J and W Electronics
Before that, check for loose wires, loose connections, loose/cracked solder joints [on the board or at the motor(s)], or a short against the frame. The adhesive foam pad that secures the board may have lost its grip or dried up. If a screw and a nut was used, they may have loosened up.
Also remove the trucks and check the worm gears. Sometimes grease cakes up in there. See whether the wheels and gears turn smoothly and easily. Maybe something worked its way in there.
Another topic mentioned brittle wires breaking on Williams illuminated passenger cars. That's another thing to check.
If the board isn't working, Williams and Dallee reverse boards and sound systems are good options, too.