I currently know that unity has the ability to support oculus rift. However, in preparation for the imminent arrival of my rift I would like to know, if the unity web player has support for the oculus rift? I have already looked into webVR and it seems like a great standard for VR on browsers in the coming future.
You talked about Web player in your question and tagged Web player and WebGL in your question. I want to begin by saying that Web player is not only discontinued but now removed from the current version(5.4.0b13) of Unity beta. Forget about Web player.
WebGL replaced it.VR is NOT currently supported in WebGL build. There is an attempt to develop WebVR standard for VR in browsers but that is new and that must be usuable before Unity will decide to support it. VR support is NOT on the Unity timeline or to do list so don't waste your time waiting for this support. Unity is working hard to support WebGL 2.0 and to fix other many WebGL bug before adding another a new feature for WebGL build.