I'm currently working on a project that uses Awesomium to render a webbrowser. Currently I'm having succes with rendering one webwindow with mouse input, but as I'm working on the project more work will be add later. You can also look into the unity-berkelium wrapper hosted on github.

Screenshot from prototype

You can find my project here

Not in any easy way.

If you target the web player, you can use some javascript code to hide the unity player, and display an html frame instead. This works, but won't let you truly embed it within your player, more like replace your player view with a web view (at least on most browsers, as so far, only Safari on OS X truly supports layering of Unity with other html content).

If you are targeting standalone or other platforms, your only options would be to integrate some browser tech like WebKit using a C++ plugin, or to write your own basic browser in C# (depending on the level of sophistication needed). Either way, it will be a lot of work to get basic functionality running.