My neighbor has a JX75 that is on it's last legs. One of the rear wheels doesn't drive, but the other still does.

What is the likely cause? Is it adjustable? Will this require a whole new transmission?

Hi,
The drive system needs servicing. Suggest you do both sides and just figure you will replace the spring and key inside the pinion. Also, the gear attached to the wheel may be stripped. Many bad things happen to that model if the back cover inboard of the wheel gets broken and allows in dirt. See drawing.
Peace,
Clarke