Does the school permit her to carry hand sanitizer? Only the school can answer the question of student-carry on campus. I suspect that the answer is/will be no due to the hyper-sensitivity schools have to being sued. Check the schools website for their student requirements handbook that delineates appropriate student conduct and prohibited items.
By definition what is not made illegal is legal but schools have long been permitted to violate the laws to ensure a "safe and nurturing learning environment" even if that entails violating every one of your child's enumerated rights.