That would take hacking skills (Knowledge of IP addresses, the specific software used to run the web cam, his firewall and virus software etc. etc.) that the school is unlikely to have access to. It would also make them guilty of hacking a computer system, also a crime, as well as invasion of privacy.
They got pictures off his webcam somehow. That means it can be done.
Other than that, what Wurlitzer said.....
I would expect the school to have the records of the things you mentioned necessary to get into the computer. And in that case, it wouldn’t really be hacking.