Gsuite may or may not be a problem - do you use the gsuite sync utility and if so, does it support color categories?
I have a round robin macro that uses categories but color categories don't work right with imap accounts. The next option would be moving the messages to a folder (each user has their own folder) - the round robin macro could be tweaked to handle that. It needs to run on one computer so sort the mail.
How to use an Outlook macro to move messages to a new folder or mailbox or to assign categories to the message round-robin style.
www.slipstick.com
A low tech option is to tell users to move the messages to their folder when they claim them.