Mail this page definitely does work with IE9 - I will be happy to send you a link on our testing site where you can see it working in IE9.
The problem is with your site template. You have a huge number of scripts on your page, one of them appears to be incompatible with mootools in IE9, which is Joomla's javascript library. If you use IE9's developer tools and try to debug your page you will see a lot of errors being flagged in mootools, this usually happens when another script re-defines function names that mootools uses, its a naming conflict.
The Mail this page module uses mootools to create the slide effect, hence it shows up with this problem.
I suggest taking this up with the template author, it looks like a poorly written commercial template.