Earlier this month, the W3C published an Editor's Draft for WebVR 1.0. The specification has not yet been ratified, but the proposal is backed by engineers from Mozilla and Google. It enables the use of VR headsets in the web browser, including all the security required, such as isolating input to a single tab (in case you need to input a password while the HMD is on your face).
Firefox Nightly, as of August 16th, now supports the draft 1.0 specification.
The browser currently supports Oculus CV1 and DK2 on Windows. It does not work with DK1, although Oculus provided backers of that KickStarter with a CV1 anyway, and it does not (yet) support the HTC Vive. It also only deals with the headset itself, not any motion controllers. I guess, if your application requires this functionality, you will need to keep working on native applications for a little while longer.