If they want to see live images free, I do believe that Yahoo Messanger, MSN Messanger, and maybe AIM now all offer it as part of their package. If they are a mac user, there is isight.
Now for webpages, you can record some video, upload, and then they can play the video by clicking on a picture or text link and have it linked to the video. It is best to use Quicktime format if possible (Quicktime Pro is only $30 for Win or Mac) since people can then play it on a mac, windows, Linux, or even PDA if they have one of those slick coloured one.
Unless you have a large family, chances are you can be safe with like 5GB of transfer a month. Webspace, I would say at least 100MB, more if you are going to place full lengthed features baby movies.
How?
make the movie, then use Quicktime (okay Windows Movie Maker if you *have* to) and compress the film into a .wmv (windows) or .mov (Quicktime). Then upload the film to your website via FTP (chances are it will be too large to do so via a browser form) in binary form and then put a link like:
http://mynewbaby.com/babymovie1.mov
in your webpage or just email that link to everyone.
Cheers.