In the source for the page there are some comments about the (default) URL for the video being defined in the file cf_introduction_config.xml. There is indeed a reference to the XML file: http://37assets.s3.amazonaws.com/video/campfirenow/cf_introduction_config.xml.
Downloading this file and inspecting it reveals two URLs:
The last file has the file extension normally used for Flash videos, flv. However attempting to download it with Wget results in a 403 error.
The first file, a SWF file (for displaying "animated" vector graphics), can be downloaded, though. It has a length of 9.5 MB. It can be converted to a working MP3 file, so at least it contains audio. I used MediaCoder to get the MP3 file. Could it be played in Gnash and show some animation?
Gnash may be used. From the FAQ, "What operating systems does Gnash run on?" there is this somewhat cryptic statement:
Windows builds are available (compiled against Mingw).
But it is not clear where such builds can be found, if they can be trusted and if they require Mingw to run.
I tried to install Mingw, but it required too much effort. It is not straightforward.
In the end I dragged the SWF file onto Firefox, and it worked! Probably because the plug-in Shockwave Flash was installed. I haven't installed it, and I don't know how it came to be there. Pretty scary. So much for security.