Double check the markup on your checkout page by viewing the source and running it through a validator. Keep an eye out for unclosed html tags and misplaced code. For example if you accidentally style the shortcode in the edit page screen, WordPress will wrap it in a span
tag, invalidating the html and causing the checkout form to not submit (without any errors) in IE9.
One example of code that will cause the checkout form to not work is:
<p><!--:fi-->