'embedded' site content

The best way I can think of to describe what I want - to display a live web page on the surface of a mesh; animations, clickable links, etc.

Something like this can be achieved by applying 3D CSS rules to an iframe element. These rules, in turn, should be calculated from the current camera position in your WebGL scene. This method has its limitations of course, namely, you cannot render a 3D object in front of the iframed website and have interactivity at the same time.
It however will work if your website surface is intended to be in the front of the scene all the time, like in the above-mentioned example.