I am DVC and we do Magical Express all the time. So I assume renters can too. The member you are renting from might have to make the request for you.
You do not need car seats on the Magical Express bus. I've never seen one in the many busses we've ridden on. If you bring it for the plane, you will probably store it under the bus. For at least the two year old, I would think about bringing it on the plane. My DD was much better when she was sitting in her car seat. She understood that she just couldn't get out and that was the way it had to be. By 4 we could amuse her with coloring books, games, etc. Now they have portable DVD players which are great on the plane if your child will wear the headphones so everyone doesn't have to listen to the movie!
The Magic Express bus will drop you off at your check in. Depending on the resort, you might want to have bell services help you to your room if you have a lot of luggage. You can gate check a double stroller if you are bringing one and that will make getting around the resort at check in much easier.
It will be several hours before you get your luggage. And depending on what time you arrive, you may not be able to get into your villa. If you don't plan to do a park on arrival day, you can bring swimsuits, etc in your carryons and change at the pool. Even if your room isn't ready, you can use the pool and stuff. Bell services will also store your carry on luggage if you do want to head out to a park. We've had great luck doing that!
Hope you have a great first trip.