Short of staff incentive schemes or threats of redundancy, how about using task scheduler to run the email program. I am not sure if this will work, but there are options to run the program daily. Then you can repeat the task every 30 minutes and run it for 29 minutes. If someone closes it, it will soon run again. I guess you could run it every 20 minutes for five minutes if that is enough to do its work.
Another solution might be to hide the running task from users. Some programs have the option to hide the system tray icon.