I emailed Sylvia Pfeiffer about the implementation status of WebVTT. When can developers use it, if not immediately? She replied that there is support (probably incomplete) in the IE10 developer build, work is underway in WebKit, and Opera's status is unknown. She recommended that web developers who need captions right away should use a polyfill such as captionator.js.
I'm reposting our conversation to make the answer Google-able for other developers looking into WebVTT, at least in the short term before this information goes stale.
What's WebVTT? Captioning for online video. See this talk by .