We have used the resort chairs on at least five occasions and stayed for a week most of those times. We had no trouble getting one, unless you count the time at BC where they called to YC to have one brought over.
We have borrowed them at CR, BC and AKL and taken them everywhere with us without any problem. Like I said we have our own vehicle so we did not use Disney buses but we did us their other transportations like the boats and monorail and had not one comment or problem.
We did sign for our chairs and like I said we left a credit card number and ID. We did not use the chairs that are kept by the valet area to help guests enter the resort. I would guess that those chairs are meant to only assist guests to enter or leave the resort, a bit like the ones neat the AK parking area that help you get from the parking lot to the wheel chair rental area only.
Perhaps these are the chairs that are not permitted into the parks?