For support reasons I want to be able for a user to take a screenshot of the current browser window as easy as possible and send it over to the server.
Any (crazy)
A webpage can't do this (or at least, I would be very surprised if it could, in any browser) but a Firefox extension can. See https://developer.mozilla.org/en/Drawing_Graphics_with_Canvas#Rendering_Web_Content_Into_A_Canvas -- when that page says "Chrome privileges" that means an extension can do it, but a web page can't.