help adding a password retrieval button to the password form

Hello, I have some pages set as private-password protected on my site. I would like to add a “forgot my password” button to the password form. The function of this button would ideally allow the user to submit their email address so I could verify they are approved users and manually send them the forgotten password.

Any ideas on how to do this? I am not familiar with php, so I have no idea how to customize the theme functions to do this.