This thread will probably be moved to the disabled cruisers thread but in the meantime:
You do not go through Princess to rent a scooter. I have rented directly from Special Needs at Sea probably 5-6 times, including last month and have always been extremely pleased with the service, and the equipment. The scooter is always waiting in the cabin when we embark, and I leave the scooter in whichever location is designated as the disembarkation waiting area for passengers needing wheelchair assistance (on our last cruise that was the Crown Grill but the location will vary). Note that we always cruise in a handicapped accessible cabin - so have never had an issue with storage or getting through the door. You must store the scooter in the cabin - there's no place to leave it overnight, for example.
You will also need to fill out a form giving Princess the details about the scooter - height, weight, etc. The form is easy to find on the website, and the scooter rental company can provide the details.
Most passengers are very considerate of scooter users as long as the user is considerate! Getting on/off elevators, for example, has never been a problem for us. You can scoot right to your table in the dining room and one of the staff can move the scooter to an out of the way location while you dine. I've never had an issue taking the scooter through the buffet, although you have to go very slowly and people will absolutely step in front of you. Common sense applies and your MIL should be fine.
If she isn't already experienced at using a scooter, I suggest some field trips to the local grocery store first, so she's comfortable with maneuvering, especially backing up. Note that any rental scooter will have a speed dial (think from turtle to rabbit speed) which will FAR exceed any grocery store scooter, sol I also recommend keeping the speed on very low until she's comfortable with it.
Enjoy your cruise!