Nope... You can take your Midol, Tylenol or whatever you use to the nurse and the nurse will dispense the medicine to your daughter.
She does NOT need to have this on her at school.
I'm sorry about her cycle being hard. My daughter (now 17) had pretty bad cramping, etc each month and when she was regular, she would take a Midol before school and then take one at school via the nurse if needed. About once every 6 months or so, her cycle was so bad and I would have to pick her up or let her stay home the first day. We've since been able to get hers under control, thank goodness.
Check your school rules information but around here you can get in trouble with any type of pills on you. ALL meds stay in the nurse's office.
Also: It sounds like you are very frustrated. Believe me, the 13 yr old phase does not last forever. Have good communication set up between you and your daughter because you'lll need that when she gets older. My 17 yr old was never "bad" and managed through the hormone swings without ruining our relationship. Just remember... this phase will pass and you'll be sending her off to college before you know it. Time flies, appreciate each moment, even the not so good ones.