Do you have any information on the private shuttle or what worked for you? Similar situation but for family of 5 with 1 wheelchair (quadriplegic). Would love to hear what you learned and / or worked out best for you please!
Again, options differ based on your equipment/ability to transfer. If the equipment user cannot transfer and/or the equipment is not "portable" (eg, breaks into pieces 50 pounds or less and/or is foldable), you need a vehicle that has a lift or ramp. Providers of these services in Orlando include accessible taxis (Mears plus a couple of others), private transfers (Mears, mostly), and buses (Mears, mostly).
If the individual can transfer to a "regular" car or SUV seat, more options are available, including ride shares, most taxis, all buses, and most private drivers (Again, in Orlando)
All, or nearly all Disney buses are wheelchair accessible. The Skyliner is accessible. Most boats are accessible. If a particular boat on a route is NOT accessible, the next one (20-30 minutes, usually) will be. Accessible transportation is a big reason for wheelchair users to stay on property.
Transportation options in other areas is different.