If everything goes right in the system input stack then the media keys on the headset should be available to applications like Chrome. There is a bug open to make them work with Chrome Apps,
crbug.com/131612, but I've yet to find something tracking making them work on websites in general. The trick is figuring out which thing on your system should be processing these events. Should they go to the music player you have running in another tab or the page you have in the foreground?
WebUSB is not a good solution for this because a) the headset's media keys are exposed through a HID interface on the USB device so you won't be able to claim the USB interface away from the OS kernel and b) as you said, headset manufacturers would have to make the device WebUSB capable.