Even with some amount of camera shake, the video will probably be able to identify the make and model and certainly color of the idiot's car. Handing the video to the police would hopefully be actionable enough that they could go through proper channels to identify the car from there.
Another thought, maybe you could sacrifice one morning sitting in a lawn chair at the entrance to the school, looking like you are absorbed in your phone, but videoing all the incoming cars/license plates. If this car drops off kids every day you should see them, and that would be helpful in tandem with shaky video.