You may be the third person ever using gamepad :-) The game uses the html5 gamepad api.
What Operating System/Browser/Joystick do you use ? Is it an analog (stick) or digital one (pad) ?
Does connecting the joystick before starting the browser or reloading the page after connecting it improve things ?
Is e.g. the menu pointer moving by itself ? Because ...
Guess 1: A joystick option turned on which causes firing of events continuously, e.g. autofire ?
Guess 2: Maybe your joystick is an analog stick and in rest position yields already an axis value which is above the fixed threshold I have chosen (0.25). I did not implement a full calibration process. Worked for my XBOX controller and logitech precision pad in tests.