First of all I guess it's possible, but you have to program a solution yourself ( I don't know any existing solution) :
To catch every client side Exception, use setUncoughtExceptionHandler. At this point I can recommend gwt-log, which is easily set up and can be used to send all client side Exceptions to your Server, where they can be logged.
But you have to combine both, because you need an event which triggers the image creation and the sending of the exception to the server. This is because you can either use setUncoughtExceptionHandler
or attach gwt-log to your client.
Alternative you can do it without gwt-log and then send both by yourself (image and exception) at the same time to the server, to store them combined.
To get a screenshot you can use html2canvas, which renders something like a screenshot out of the DOM.
All in all I expect it will take some time to set up and programm all parts of the solution, but in the end it should work.