This is probably a bit weird, and I don't think it exists yet.

I'm basically asking for a program that, upon receiving a request via email, downloads a html page and sends it via email, possibly changing the links inside the page into outgoing emails to this program asking for another page.

This is certainly one of the most crude ways of accessing the web, and obviously fails at anything beyond the most basic stuff. But it may still be useful to those that can send email, but can't access the web due to company policy or whatever reason.

In the (likely) case this does not exist, I'd be interested to write such a proxy.