The problem is that without the cron, you can't schedule a script to look for email every X minutes. And without shell access, you can't tell your server to re-route email to a PHP script.
That means you need some other way of triggering your PHP script to parse the email.
One way (one bad bad naughty bad way) would be to start a script that 'never ends'. All it would do is check for email, go to sleep for 5 minutes, check email again, sleep 5 minutes, etc..
Problems:
- if it ever crashes, it won't re-start automatically
- it ties up system-resources (not polite on a non-dedicated server)
If I were you I'd contact the system administrator to see if you can get him to
let you use the crontab, or better yet, get him to setup an email reroute to a PHP script.