Hi there, Patricia!
You are such a smart thinker to plan for this on vacation. I visited
Walt Disney World once not long after having surgery, and it was tough trying to get comfortable in bed after sleeping in a recliner for a few weeks.
As you've found, there are no recliners in the rooms or available through
Disney's Grand Floridian Resort and Spa. You do have to rent from those outside companies. You are permitted to have them in your room at the resort, however, you must contact the resort prior to your arrival to let them know that there will be a chair delivered. There may be a need to adjust the configuration of some of the furniture, or there may be a possibility that something would need to be removed from the room to make sure you can accommodate your recliner.
"Mousekeeping" is also able to help you with things like extra pillows, an additional set of linens, or whatever else they might be able to provide to help make you more comfortable. I have always found the cast members to be very helpful in situations such as these.
I hope you get a good night's sleep, have a wonderful vacation, and make many magical memories!
Anna