It appears I broke the ticketing system. I disabled autocron, then put the following entry in crontab:
* * * * * nobody /etc/php5/apache2 /var/www/support/api/cron.php
When I check the logs using sudo grep cron /var/log/syslog, I can see that the job is running every minute, but my test ticket is not showing up, even when I refresh the tickets page. I'm guessing I've specified the cron job incorrectly, and since autocron is not running, OSTicket has no way of knowing when it has new tickets.
Do I have to specify the php.ini file, or am I on the wrong path entirely?
Thanks again for your help.