The line was missing, so I added it and restarted Postfix. Filled in the form and the user has still received nothing.
We have tried this using an email address not on the server, so we know the form works and sends email normally.
We have tried sending emails using the user account but just with email clients, not from the web site. That works, too.
It is definitely a problem sending mail from a web site on the server, whether to the user's email client or to a forwarding address.