In general it is not possible, because your system can replace the font face specified by CSS with something else, and the browser doesn't know. For example, what font is used for "font-family: serif"? Fontface Ninja doesn't know.
In these cases you can sample the font and analyze it with something like WhatTheFont! or Font Matcherator.
https://www.myfonts.com/WhatTheFont/
https://www.fontspring.com/matcherator