The type of input device on the client end shouldn't make any difference. The browser accepts the input from the OS, which get's it from the hardware. The client code takes that click/touch information and translates it to the board.
I mean, it is potentially possible that the code is interacting in a weird way with the inputs but I would lean towards a screen calibration thing. How do you make your moves? Do you try drag and drop or touch the piece and then touch the destination square?
I have a touch screen laptop and when i started using it to play .com i was really excited becuase i could just use the touch screen to move the pieces. But for some reason that i cant seem to fix it wont let me do that anymore, even though all other touch screen things wont on the laptop, making me think its something with .com