I wrote and used a minimalistic one to save me time: plain e-mails to people with "[t]" in the subject got put in a database, recipients got modified e-mails with URLs to a web interface in the body (in addition to the original e-mail). When the task was completed, you only needed to click on one link in the web interface (or in the e-mail) and the original sender would get a confirmation by e-mail (i.e. you didn't need to write that e-mail yourself). The web interface provided an overview of open tickets and statistics. Later, comments, re-assigning tickets, priorities and other stuff was added.

This was used for > 10 years (for a few 10000 tickets) AFAIR until it got replaced by Jira. It worked well and was accepted (or so it seemed) because it didn't really interfere with the typical way of doing things till then (e-mails) and offered minimalistic features on top.