I was having a similar problem in GMS 1.4, not sure how much help it'll be for 2.0 but worth a look.
For reference GMS 1.4 HTML5 Export was (and still is) returning gamepad_get_device_count(); = 0 but I'm now playing it with a gamepad.
If you need it for one gamepad then it looks like the HTML5 export will still let you use the gamepad if you use device 0.
So if you're checking gamepad_get_device_count() to start your gamepad code try gamepad_is_connected(0). That worked for me. Could be that checking specific slots works fine? Haven't checked but you could just loop a reasonable number of devices since the number of slots available will never realistically be higher than 20.
Also found that it gets some buttons mixed up? Could just be the controller (DS4) as I only have one to test with but the triggers swap with the right thumbstick.
html5gamepad.com/ revealed that one for me.
Hopefully that helps, very little information out there about this.