I don't get why you'd need a physical robot arm? Just have a usb device that simulates the mouse movements. Or just software to move the mouse virtually.
You don’t, you wouldn’t even need a webcam, you could just splice into the video signal at output and interpret it directly (much as the HDCP bypasses do).
I can see it being possible - racks of "robot arms" that move mice based on whatever criteria is needed (in this case, reCaptcha).
It works for 3D printing, as well as device testing - so in theory, this could be done too...