I was looking around for free fonts usable as webfonts that would allow me to put chess games on my web site in Figurine Algebraic notation.

There are some fonts suitable to that purpose, but as these web sites evidence, they're not being used as web fonts just yet...

http://mip.noekeon.org/HTMLTTChess/index.html

http://www.strategems.org/fonts/fonts.htm

instead, users are still expected to download the special fonts and install them on their computers.

At least with commercial fonts, apparently some sites are using webfonts for Chess,

http://www.dnaindia.com/sport/commentary_live-wcc-2012-anand-appears-to-have-weathered-the-storm_1687859

although it's possible that the site is just using something that I happen to have on my computer. (The white Knight, for example, is the letter N: the font is not Unicode, so it's easier to compose the chess games without special software.)