If you want to connect your NES Classic Mini's controller to your laptop, you are out of luck. The only adapter I heard of is not produced (you will have to order electronic parts and solder them yourself, without any warranty.)
The game, be it on desktop or on web version should however be able to work with any USB controller. You will want one with a good D-Pad cross for the best experience. Xbox One's controller works well, 8BitDo controllers are also really good.