What self-hosted help desk would work for an air-gapped network?

I'd start with GLPI and Zammad. Both came up as help desk options, so they're worth putting on a shortlist alongside your actual workflow requirements. Test deployment and updates without internet access before committing to either.

Request Tracker is worth a look if email is central to how your help desk works. It can be tricky to troubleshoot, though, and moving away from it later may be rough. I'd test an export and migration path early rather than finding out years down the road.

If your team likes the project and issue tracking side of Azure DevOps, consider YouTrack or Redmine. Redmine also has a wiki and project management features, though you don't have to use those just to track tickets.

ServiceDesk Plus and Halo are other products to evaluate if you want a more complete service desk. Check the current self-hosted licensing and offline requirements with the vendor, including how updates and support work on an isolated network.

Watch the maintenance burden too. A large enterprise platform may need more specialized administration than a straightforward ticket queue, so match the tool to the processes and staff you actually have.

For a simpler queue, HESK is another option and was suggested as lightweight. Jitbit is worth checking too, though verify its current operating system requirements before assuming it'll fit your servers.