Billy doesn't drive, Jacob drives him around. In chapter twelve of Twilight, Billy and Jacob visit Charlie and Bella. When they get out of the car, Jacob had been driving Billy around. Charlie, being a police officer, comments on the fact that Jacob, who is underage, is driving. Jacob counters with a lie that they get permits early on the reservation. Charlie lets it go, because Billy says, "I have to get around somehow." Since Billy is in a wheelchair, Jacob drives him around, even though Jacob doesn't officially have his driver's license yet. 

There are any number of modifications that allow someone in a wheelchair to drive a car.  The car or van can be changed so that all the necessary things are automated so that the disabled person is able to drive.  Because you can be in a wheelchair but still have a full control of your hands, etc., the car can be changed so that there is a hand throttle, ramped entrance, etc.  If you watch Lost, there are episodes where Locke shows his prowess getting in and out of the car on his own despite being in a wheelchair.  Thanks to advances in technology, there are more and more ways for those having to deal with specific disabilities to be able to participate in the rest of the world without as many limitations.