There is a lot of walking at Walt Disney World no matter what transportation you take to get to the parks, I’ve walked 5-8 miles per day. If your mother would be willing to rent an ECV it might make your vacation a little more comfortable for her. The buses can easily load an ECV and then she would have it in the parks and at Coronado Springs (which could also be a lot of walking depending on your room location–be sure to book preferred or request to be close to the main building).
Taking the bus will definitely be more convenient at the Magic Kingdom. If you get there very early (before the park opens) you can generally park in the first several rows. Each park has a parking lot tram that minimizes walking if you drive to the park and have to park far away. Renting the car will probably get you back to the resort faster than waiting on Disney transportation, but you will have to walk farther than you would from the bus drop off points. If minimizing walking is more important I would ride the bus; if getting back quickly for the nap is more important then I would rent a car.