I have a client that I have designed a site for. On the contact page, I have a feedback form. The client is requesting that I add an option to the form that the site visitor can add an attachment from their computer to the form.

Can someone help me out here?

not with HTML.

It will need sever side code to upload and attach the file

The following page contains sample code(requires PHP):
php upload form

assuming that the OPs server supports PHP of course.