Can a NY high school search a student’s car without a warrant? ?

At my small suburban high school in Central NY (Syracuse), there is a state trooper who has an office in the building and is here at nearly all times during the school day. I know that lockers and bags can be searched, but: If a student was suspected of having something that they should not in his or her vehicle, can it be searched? And what is the procedure by which the searching would occur? Would the student be notified beforehand?