Wow! I just looked at ABBF's website and they have added a few items!
I would think that you could contact ABBF and ask them about Disney policies concerning refrigerators at the resorts. I can't believe ABBF would purchase them without being able to rent them out to guests at WDW resorts. They certainly are cheaper from ABBF if you need one longer than 3 days.
I called. She said it is delivered to the luggage room before we arrive (just like everything else we rent from them), and is in a plain box wrapped in a plastic wrap. She said no problems so far, and I do not anticipate any problems with my resort.
We have ordered from abbf before and have be very pleased with everything. They are highly recommended.
We brought our own and noone ever batted an eye, so I can't see there being any problem with you getting one from ABBF. If they allow us to bring one, then they can't see you can't just because you rented yours.