Looks like this works on Chrome for Android, but Firefox doesn't seem to support HDR at all.
https://bugzil.la/hdr
Maybe some day.
Neither does Safari on macOS – which honestly seems like the correct behavior, given that this will inevitably be used by websites in user-hostile ways.
Safari only shows HDR in videos, not photos. It's possible to just show a single-frame video though.
And I could see Safari changing its behavior to only allow HDR for actually playing videos (or even playing videos with actually varying content, if websites start playing clever games with one second loops of still images), or maybe only after confirming an "I am wearing sunglasses right now" prompt.