Is it possible to create a mailing list function without PHP / database software?

Mailing lists are an important part of many websites, allowing site admins to issue notifications and announcements to the users. My question is, would it be possible to create a mailing list system without storing the emails using PHP and server storage software (eg. MySQL)? To be more specific, I'm looking for a script that, when the user enters their email into the mailing list form and clicks "Sign me up", the user's email is sent to another email to be stored in the inbox of that account, instead of on the site's server.

As an example, the email "msmith@gmail.com" is entered and submitted via a mailing list form. The email is sent as a message to an account controlled by the original site – let's call it "mailingListUsers@examplesite.com". Then, it could be copied down onto a database for further use by the site admin in control of the mailing list.

Is this possible with current web development tools? If so, what is the code that would allow me to implicate this function into my website?