Yes, equipment rented from an outside vendor can park-hop with you, go to Disney Springs, visit resorts, etc. The challenge may be the 1-day timeframe, as typically the outside vendors rent for a multi-day period. You'll need to discuss that to see if the vendor is willing/able to work with it. Since Walkers is allowed to access the PC port terminal pre- and post-cruise for rentals, they may be able to work something out. If using Special Needs at Sea you would probably rent for the duration of the cruise, with pick-up/return via the NYC terminal. There are other vendors who work just within the Orlando area (not PC area), but whether you could arrange a 1-day rental with pick-up/return at a park is questionable.
Enjoy your cruise!