FormMail doesn't insist on having the sender's email address. If present, the email field is included in the "From" line in the message.
The first step is to use the test mail script you can find here: http://www.tectite.com/vbforums/showthread.php?t=62
It doesn't include a "From" line, so if it can send email on your server then FormMail will be able to send it too without the email field being filled in.
There could be a couple of reasons why it's not working, so check if the test email script works, and we'll move forward from there.