We're in the same situation. We've found that it's easiest just to start the trip in the wheelchair and use it the whole time. Just walking from the room to the food court a few times would probably be too much walking for DH, and it would take him the next three days to recover from it. In other words, he gets into the wheelchair before he has to, because by the time he needed it, it would be too late for him to have any enjoyment of the rest of the trip.
The Disney resorts have wheelchairs that you can borrow, with a deposit, for your entire stay, but they can't guarantee one will be available on arrival. We've started taking one from home (we got it because it's useful for anything that requires a lot of walking, like trips to the mall, etc.) It's beyond convenient to not have to worry about renting one at the parks, and the Disney transportation system makes it very easy to get around with the wheelchair as well.
On my first trip to WDW in 1980, when I was 8, my grandmother broke her leg three weeks before our trip, so she needed a wheelchair. I am constantly amazed at how far WDW has come in improving accessibility in the last 25 years. Our experiences with the wheelchair now are completely different.