Yes, there is transport. The Disney drivers will pick up inside the campground, just like the regular campground transportation. They stop at each bus stop in the CG. Then they take you to the Wilderness Lodge where you meet up with the big bus that will take you to the start line.
It sounds inefficient, but it works really well. Our Disney driver told us that they do it this way because they tried letting the contract companies drive through the campground, but they kept getting lost or stuck in the tight spots the Disney drivers know to avoid. So it was easier to let the regular drivers do the CG and have them meet up with the contracted transportation elsewhere.