Posting just in case someone else has this problem, or I forget how I solved it.

I'd been trying this for a while with no success, the command ran fine from the command line, but in cron it just did not work. No errors, just did not refresh the calendar.

Original crontab:
05,20,35,50 * * * * zmmailbox -a admin@zimbra.site.org -p pfffte -m wiki@zimbra.site.org syncFolder "/TicketSystem"

Working crontab:
05,20,35,50 * * * * bash -l zmmailbox -a admin@zimbra.site.org -p pfffte -m wiki@zimbra.site.org syncFolder "/TicketSystem"

For some reason zmmailbox needed a login shell. Evironment vars??