If you are getting copies of the emails, I would instead post correct info somewhere "rebutting" everything without rebutting.
Their email: Shitty, defamatory bullshit.
Your posts somewhere: Factual, complete, respectful, never mention that it's a rebuttal.
Also, track everything, search up local laws, find some way to notify the perpetrator that their criminal count is rising:
1. Theft of the email list.
2. Violation of any legal obligation they have to respect student privacy, non disclosure agreements, etc.
3. Defamation which may not get them charged criminally but could get them sued.
4. Multiple counts of defamation and libel as each email tells lies about presumably different people.
5. If there is anything remotely sexual in any of it, go over all laws pertinent to sex offenses.
Try to find some means to notify parents they should not trust the info from this source, you are doing what you can to stop it, each email sent likely adds counts to what they can be charged with and you need help tracking it for legal purposes.
A lot of people get told to delete offensive emails from stalkers for their mental health. It de facto destroys evidence and helps cover up the crimes.
Provide people a list of best practices for dealing with social aspects of the issue. If an email says "Sally is a slut!" do not argue about Sally and her sex life. The correct thing to do is remind people this comes from a malicious actor, it's untrustworthy information intended to hurt Sally and her sex life is no one's business.
Rinse and repeat: "This is intended to be malicious. The source is a thief who is likely a disgruntled employee. Their actions amount to career suicide. It's only a matter of time before we identify them. Until then, any information from this channel should be disregarded."
Less personal info that is more relevant to school function:
Perp: "The school is pissing away money for stupid reasons!"
You: "This is our budget. This is why it gets spent that way. Sad that things are so expensive but doing x instead of y fails to achieve the desired result."
Real world example: A school in a hot, humid place turned off AC all summer while school was not in session to try to save money and came back to a rampant mold problem that they couldn't afford to remediate. Turning off the AC was penny wise and pound foolish.
As much as possible, do not mention the malicious email campaign when publishing accurate information to keep people informed about what's real.
When it's over, assess what parts of this information campaign you want to continue with because it was unexpectedly beneficial to inform people.
Best of luck.