Hello, Megan!
Breakfast at Be Our Guest Restaurant sounds like the perfect way to start off your day at the Magic Kingdom. Typically, Guests rent
Electric Conveyance Vehicles (ECV) once the theme park opens. The rental location is just past the Mickey touchpoint admission area, so you may find they will open a little earlier than the theme park. However, there is no guarantee and may not provide a solution for your dad prior to your breakfast reservation. Your dad may benefit from renting a unit from a Disney featured provider so he can have the freedom to use the rental throughout the Walt Disney World Resort. You can find a list of Disney featured providers at the bottom of ECV Rental page linked above.
My mom uses an ECV to tour the resort, and we have had great experiences using Buena Vista Scooter Rentals. They offer a variety of ECV models, reasonable prices and will even deliver the unit to most Disney hotels. We pick up the ECV at Bell Services when we check-in and return it there on our departure day. It is convenient and makes touring the theme parks easy for my mom.
I hope this helps, and you make many wonderful memories during your special breakfast. Please let me know if I can be of further assistance.
Missy