I use a couple year old JVC with a firewire output and produce great streaming video with it. Anything though with a composite out or firewire will work. Composite out (line out) will require a USB "dongle" or a Dazzle-type device to digitize the composite signal.
Unless a camcorder has a webcam option, don't plan on using it as a streaming camera through direct USB. Most hard drive cameras have a direct USB but that's only good for transferring files from recorded material.
If I were you, depending on the amount you want to spend, I would hit up a pawn shop for an older JVC or Sony product.
As for a webcam, make sure to buy a good one, but not too good. If you start to get into more expensive webcams with special features, like zoom, you will likely have a less than stellar streaming experience. A medium-grade $50 webcam works fairly well but you should not expect any depth of color and adjusting overall contrast can be difficult. I would personally stick with the video camera option for the simple reason that you can record in great quality at the same time you stream.
Hope this helps.