Folding seats or chairs are listed on the Disney website as something that is not allowed to be brought into the park.
Occasionally, you will see someone with the type of folding cane that has a small fold down seat attached to it. I would not count on being able to bring one of those in because it is really a folding seat and I have heard of people who were told they would not be able to use it in the parks.
We were just at Toy Story Mania the day before yesterday and this is what happened to us when DH, DD using a wheelchair and I rode.
At the time we got to the last part of the ramp, which is in the accessible boarding area, we were asked how many were in our party and whether DD could transfer out of her wheelchair.
Everyone else in the line was asked the same thing as they made the turn or got near to the boarding area. The CMs did not leave the actual boarding area to talk to people who had not passed the last turn.
This has happened every previous time we have ridden Toy Story Mania, from the first time we rode after it opened.
The 2 groups immediately ahead of us were:
1) a woman in a power wheelchair who did not transfer, a male and 2 small children
2) 2 people using ECVs who planned to transfer, 2 adults and 4 small children.
When group 1 was at the front, the CMs transformed the ride car so she could ride in the wheelchair. The man and the 2 small children rode in the front row of the ride car in the pod behind her. We were asked to move all the way to the right side of the queue so that a group of 2 (one of whom had an ECV and was transferring) could get by us and fill the last row of that ride car.
Group 2 was told that our group would be coming in front of them when the wheelchair accessible ride car came back around so that the CMs would not have to convert it back to a seat and then back to wheelchair accessible for us to ride. They were told that a 'regular' ride car would be called to the accessible boarding area for them so that each child would have a seat and no one would have to ride on laps.
Group 2 was pulled forward into the waiting area in the photo I posted and we came past them to board when the accessible car came around. A party of 2 (one of them had a cane and could not do steps) was also pulled ahead to fill out our ride car.
This happens frequently at places that have wheelchair accessible ride cars. They do try to maximize the number of people loaded, but there is no guarantee that someone who can or can't transfer will be getting on more quickly. it's often random, based on the needs of the other people in the accessible boarding area and the numbers of people in each party.
For folding chairs, the website does specifically state that folding cane chairs and walker chairs that are primarily used as mobility devices are allowed (at least Disneyland's does, I would assume Disney World is the same), that being said, I know many find it easier to ask to borrow a wheelchair at each ride that it is needed, especially if they might not need the seat at too many of the attractions.