It would be a security hole to allow screen shots to be retrieved by any website, so that is definitely not going to be allowed. But even if you could force the browser to press Ctrl+PrtScn, it would be OS-specific and you still wouldn't have a way to retrieve the contents of the clipboard without the user's interaction.
There are <canvas>
libraries that can help simulate what you want, but there isn't a way to retrieve pixel-for-pixel exactly what is open on the user's screen.
http://html2canvas.hertzen.com/
This script allows you to take "screenshots" of webpages or parts of
it, directly on the users browser. The screenshot is based on the DOM
and as such may not be 100% accurate to the real representation as it
does not make an actual screenshot, but builds the screenshot based on
the information available on the page.