The exact same HTML Validator extension/code works fine in Chrome. The reason being several bugs and mostly one API not implemented in FF.

This is the chrome version (in fact the same file): https://chrome.google.com/webstore/detail/html-validator/mpbelhhnfhfjnaehkcnnaknldmnocglk

For more explanation, please look in the bugs below.
https://github.com/mgueury/html_validator/issues/11
https://bugzilla.mozilla.org/show_bug.cgi?id=1361121

and mostly vote for the FF bug to be fixed :-)

Thanks,