Zammad provides web-based ticketing that can be self-hosted. The Zammad alternatives entry is most useful with a realistic handoff rather than a ticket created and closed by one person.
Let one test agent receive a fictional request, add an internal note and pass it to another agent. Check whether the second person can reconstruct the issue without exposing internal discussion to the requester. If self-hosting, include backups, updates and mail delivery in the operating plan.
Akousa