So i know the post is old, but it was a recent problem for me and i had to do some serious digging and thinking outside the box.
basically like most replies here - you cannot alter the webbrowser control to use the most recent IE engine. mine uses IE7 by default, i have seen some replies that basically changes/ adds stuff to registry, am always not comfy when it comes to the registry, a cleaner way to address this issue would be to append code on your website that forces it to use the most current IE engine on any pc and works like a charm.
if you have access to the web.config file of the page you intend to browse, simple append:
<system.webServer>
<httpProtocol>
<customHeaders>
<clear />
<add name="X-UA-Compatible" value="IE=edge" />
</customHeaders>
</httpProtocol>
</system.webServer>
and your site would force webbrowser control to run the most current IE engine on your computer. Other options are found here:
https://www.leapinggorilla.com/Blog/Read/1016/ie-ate-my-css---disabling-compatability-mode
I should state that this would only work if you have access to the page / web.config of the website/ application you are trying to access- which was my case.