Dalek, my first idea was using VLC which I knew was a streaming software, but I was not sure it would accept camera input. I used VLC for different purposes like downloading/conversion in tricky situations, not streaming. Important thing: this is Open Source.
See also http://en.wikipedia.org/wiki/VLC_media_player
] and http://en.wikipedia.org/wiki/VideoLAN
]. Originally, VLC means VideoLAN client.
(No cameras do not come with streaming software, John, I used good number of them (more of industrial class though)). Some comes with embedded server or course, but this is firmware, you cannot serve it from a regular Web site.
Combined search on Goodle gave me this:http://www.wikihow.com/Stream-Your-Webcam
(By the way: a credit to Espen Harlinn: he reminded me about VLC in some completely irrelevant context, so I immediately remembered Dalek's question and decided to find out if VLC could serve up a camera.)
Please see if it's good enough for you.
If you need something better, try to Google "stream camera to web" or "stream camera to web VLC".
Please share you experience if you find and implement something interesting.