Host your own opensource with Drupal Burndown agile/kanban, though self-host will not suit all
https://www.drupal.org/project/burndown - this provides a Kanban or Agile Board ticketing system.
1. Get a digitalocean vps for about 10-20 pounds per month. Or host on a raspberry pi. Rationale for these smallish servers is that if the ticketing system is only used by the business, then a large scale server for a public site is not necessary.
2. Map to a web domain, if hosted on a raspberry pi over a non-static IP, use a registrar with a dynamic dns server, like joker
3. Setup a LAMP stack on the server
4. on the server get and install drupal open source CMS, and add the burndown module, also open source, free
5. setup email to send notifications, perhaps use zoho mail - low cost
6. manage backups yourself and security updates
No affiliation, financial incentive or referall benefits of the above to me. There are likely other similar open source solutions out there. Bugzilla comes to mind but I'm not sure how well it works for most recent processes these days, having not used it for many years.
The above will not suit all. There is always going to be a cost and risk whatever approach and that includes consider self-hosting. With the greater control of self-hosting, comes a requirement to have the knowledge in-house and manage the server which means billable time out of one or more staff, along with a process for absence and when those staff move on, offboarding, onboarding.
Cloud has the benefits to those whose business core competence/reason for existing is not to manage a ticketing system. Which makes the cloud a valid consideration. I've used cloud services includingJIRA in some roles as a developer, as well as trello and pivotaltracker.
Still, having options is healthy and the self-hosting may suit some who already have teams in the business of doing that kind of work already with the related skillset.