Typically, each regional competiton has a director, and you should contact him/her, If your at a university venue, you should start talking to their networking infrastructure group ASAP.
It *might* be possible to webcast over WiFi to a ethernet backbone, but this needs to be tested before the big day. As far as equipment, Kevin was right on. An old pentium III with an Haupauge WinTV Card and Real Producer (8 or 9 - free version) will work just fine.

The other caveat is to make sure you can handle the audio feed as well. The audio is more important than the video. You don't need a deck, but you may need some weird adapter jacks to get wired up correctly. Usually the AV crew recording the competition is highly professional and will be delighted to help you - often they get a kick out of webcasting their work and call all their buddies to watch what they're doing that day!