Yes, the Web Audio API (along with AJAX or Websockets) can be used for streaming.
Basically, you pull down (or send, in the case of Websockets) some chunks of n length. Then you decode them with the Web Audio API and queue them up to be played, one after the other.
Because the Web Audio API has high-precision timing, you won’t hear any “seams” between the playback of each buffer if you do the scheduling correctly.