Use the idea of "Nudge" to direct your users
"Nudge" means to make what you want your users to do the default, and require them to consciously change things to deviate from that.
For example, when you email the customers, make the return address be "no-reply," or a dedicated feedback address, though no-reply generally works better. Then have a note at the bottom of the message that says not to reply, but to go to the website to post feedback, and have a link to the form right there. If someone really needs to email you, that's fine, but put the means to do that on the website, either as a form or a mailto: link, with mailto: going to an email address used only for that link. This will make your email management easier without any inconvenience to the user.