"I'd like to set up our Helpdesk web page so people could fill out a help request form, most likely a PHP page, and submit it as a task (as opposed to an email) to the helpdesk. Is there a way to do this, or am I just a crazy dreamer?"
And here are the responses from Tommy Mikkelsen and Wolfgang Schreiber ...
If we're talking about Windows, your PHP could spawn GwSend.exe, where all info can be passed on the command line to create mail, tasks, appointments, with plenty of bells and whistles ...