Recommend a WP tool to manage a process

At our school, we want to manage the shop requests with a WP solution.

Overall, it would be similar to a help ticket system with the ability to add rules and roles.

Overview:
- Staff: Any staff can make a request. Status is "Requested." Each staff member can see all of his/her requests, the changing status, and other details for staff.
- Shop manager: Reviews request and adds urgency level, date due, and any other private notes to the Team Member. Assigns it to the best suited team member with availability. Status set to "Assigned." Needs to see overview of tickets and can sort by status, dates (requested/due), department, etc. The Shop manager and the Administrator can see all the fields; the Administrator can change which fields are visible to other roles from the back end. No other roles access the back-end.
- Team Member: Reviews open tickets. Changes status to "Working" and adds any notes. Changes status to "Done" to signal the job is ready to be checked.

And so on...

What would you recommend to check out?

Thanks in advance!