It depends on the type of scooter you were riding when you suffered injuries. New York requires all drivers to carry a minimum of $50,000 in personal injury protection (PIP) coverage when they register a vehicle. That coverage extends to drivers when they are walking or riding a bicycle but does not extend to them if they are riding a motorcycle.
Those riding Class A scooters must have a motorcycle endorsement, so for all practical purposes, insurance companies consider those scooters as motorcycles. Your personal insurance policy will probably not kick in after such an accident.
However, Class B and C scooters operate at lower speeds and include electric scooters and electric-assisted bicycles. It's likely that your PIP insurance will kick in after an accident with a Class B or C scooter; however, you should closely review your policy to make sure.