As stated, the golf carts are either off-site or WDW owned rentals. If you are staying at CR, you have a few options:
1. Boat to FW.
2. Go to MK and take a bus to WL and then take a bus to FW. There is a bus that goes between the two resorts (only one I know of that is resort-to-resort). It drops off at FW at Settlement Depot (near Pioneer Hall).
3. Drive to FW. Park and take the Orange bus from the Outpost Depot to Settlement Depot. Purple and Yellow would work as well, but take you either through the Cabins or the Campgrounds meaning more stops.
For some reason, I am thinking that there was also a bus from MK to Settlement Depot at FW, but I cannot fully remember. I cannot find any of my FW papers from my last stay in the cabins in May 2010 so I cannot check. If so, that would have been another possibility instead of the bus from MK to WL.
If by chance you see someone in a golf cart that is not full, they may be willing to drive your mother from the golf cart parking near the marina to the golf cart parking near Pioneer Hall. They are not allowed to drive the golf cart up in front of Pioneer Hall.