Grandad, it sounds like they have been served with a Notice to Quit. There is no requirement on the landlord to give any reason as no reason is necessary. IAs you say, S.1 Allotment Act 1922 (as ammended by S.1 Allotments Act 1950) allows the landlord to issue a Noitce to Quit.
The notice must be served under the terms of their tenancy agreement, it's not a statutory notice.
I understand that a committee manage the site, but do they manage it for a council?
How is the management committee constituted? Is it a company, like an Industrial and Provident company, or is it just an unincorporated association. Who exactly is the landlord?
There's no harm in asking the landlord why their tenancy is being terminated. And if it's a Council (or indeed a committee managing the site for a council) you can always ask for copies of all documents relating to the issue under the Freedom of information Act as that might throw some light on things.
As Davy says, there is no obligation to leave when the notice expires. They're completely within their rights to stay put and let the landlord apply to the County Court for a possession order. There are quite a few technical errors the council can make in serving the Notice to Quit and any one of them will make the notice unenforceable, then there are several arguments the tenants can make to convince the judge to refuse to grant possession.
Your son must find his tenancy agreement. Can you post it here so I can see what it says. Can you also post the Notice to Quit.