Using regular HTML/JS, it's currently only possible in Google Chrome.
See here for details:
http://stackoverflow.com/questions/6219197/how-to-manage-image-pasting-from-clipboard-in-html5
There are some Java applets that could work (I haven't tried any). Examples here:
http://stackoverflow.com/questions/361324/pasting-an-image-from-clipboard-to-a-website