20 years ago, I started working for schools.
I was amazed even then that they were still using fax.
One of the my early projects for any school I worked for was to eliminate it. Mostly because, by then, almost all faxes received were spam anyway. High consumable cost. Plus telephony costs. Plus there were viable alternatives just sitting there.
I spent many years going to boot sales and snapping up "real" modems (serial, not Winmodem). I still have a stash. I would then connect those modems to a machine and install something like hylafax.
Now they could fax direct from the program by just printing to a network-wide printer. All users could do it (with controls to ensure only those who should could use it). All programs supported it. And it all went out from one number so it could be monitored.
All incoming fax automatically dropped into an email account as a PDF and were distributed that way. Users who "needed to sign" something could even fax direct from the scanners / copiers if they so desired.
In every school I ever did this, actual fax usage was shown to be absolutely minimal. Spam faxes were treated like spam email. The only faxes going out were few, and they got even fewer as time went on. By the time that everyone had automated banking and communications (i.e. no longer accepting or printing cheques, school parents getting reports online etc.) fax was dead and gone.
My current place has a fax line that works this exact way. I couldn't even tell you the last time we needed to fax. The technician I tasked to set up and manage the current fax system left over a year ago. It's still working, but it honestly doesn't receive anything worthwhile at all and never sends out. We keep a single analogue line up as an emergency backup for a SIP trunk, and that operates as a fax line when it's not an emergency. It's not even worth a line on its own.
That there are modern organisations with fax still based on physical fax machines churning out paper (even MFP's), I find unfathomable. If primary schools ditched the technology 20 years ago, you should have as well. At best an all-electronic system is an acceptable substitute, but that's literally lost in the error margins of any telecommunications contract - one fax line per site, with a box to manage it but to be honest most telecoms devices nowadays just let you nominate a line as fax-to-email and you're done. Even the big SIP-trunk people have that kind of functionality.
Anything you're faxing is going to be a patient record of some kind, or a legal document (the only reason for retaining a fax in many workplaces was literally "because the solicitors say we must fax it" - but email has far taken over in that respect). Thus it needs to be monitored, stored, and accessed appropriately, not churning out on a bit of paper in an office, never to be put back into someone's records or destroyed.
They should have just turned all the lines off. Literally, if the switchboard detects fax tones on a non-fax-to-email line, it drops the call. Any analogue telephone lines should be cut and - at best - centralised to use the core switchboard as the only endpoint (so you can still make calls through them, but there's no possibility of sending stuff outside just because you snuck a fax machine in). That should have happened 10-15 years ago. VoIP is SIGNIFICANTLY cheaper that it should have been done years ago on a cost basis alone.
They are just asking for trouble, and their deadline is 20 years too late as it is. Zero tolerance should be applied. At worst, these are official communiques and need to be logged, audited and searchable in such a large organisation.
If your switchboard can't handle it, there are boxes designed expressly to convert such devices. I have a bunch of Hylafax-compatible modems if you want to do it the cheapest way possible. Most cost me £1 each.