I've been going through the Unity thread on their support of WebVR.
Here's the relevant pieces:
- WebGL exists now, but WebVR is taking a while to implement
Regarding controller support: original
It appears that Oculus Rift and gamepad controls work.
Regarding timing of WebVR support in Unity 5.7 - original post
It will not be in 2017.1 and probably also not in 2017.2.
Here's an article from Anthony Palma, the CEO of Jump on building a game with Unity using the WebVR hack.
The article above references this GitHub repository for a WebVR template using the hack.