No I think the ability for the 360 to simultaneously take the data that's being transmitted to your t.v., send it through the internet to a specific server, have it linked to your gamertag, and have it viewable via a web application would essentially be far too much for someone's internet connection or 360 to handle at all. Something like that might be possible on something like a non resource intensive XBLA or PSN title but not something like a twitch reflex run and gun FPS.
Plus the feature would be highly known all around, as well as it would have to be programmed specifically in the game.
That doesn't even include stuff like privacy rights possibly infringed which would cause a huge controversy more than likely.
Your best bet is to get a video capture card, hook it up to a pc, and have it stream over something like ustream.tv or stickam, or record the feed and just post it up on youtube or blip.tv