Cool - problem solved already, so I'll post the solution in case anyone else was wondering how to do this.
In your PHP's mail() command line, you can add a $header variable.
Code:
$header = "From: Feedback User <companyname@company.net>";
Then include it in your mail command.
Code:
mail($toaddress, $subject, $mailcontent, $header);
This way, when you co-worker opens her email, in Outlook for example (outlook...blech), she will see "Feedback User" displayed as the sender.
I don't know why it wasn't working for me without the <> part. Well, at least it works.