Hi!
I've only stayed in Good Neighbor hotels (near Disney Springs) and they have an arrangement with a shuttle company too. From what I remember most recently (June 2022) there was an accessible shuttle available, but you had to reserve it specifically. At the time, you had to make a reservation no matter if it was accessible or not, so hopefully it's just as timely. I'm sorry I don't know specifically for the hotel you mention, but this was true nearby.
Take care!
Beth