I think that if you go through WDW, they'll only arrange for you to be on a Mears shuttle. That's not what you're looking for, as you'll ride with other people and possibly make multiple stops before you get to your hotel. I don't even know if they have seat belts.
Most car companies will provide a carseat if you ask for one. Just ask when you make the ressie, and then call a few days before you arrive to confirm and make sure they have the carseat noted in the reservation.
However, I'd strongly recommend that you take your carseat with you on the plane. You didn't mention how old the child is, but s/he will be much safer in a carseat while flying. Even if the child is a baby and you didn't get a separate seat, you should take the carseat so that you can use it if there's an empty seat on the plane. If you hit turbulence on the flight, your child will be much, much safer in a carseat than in your arms.