I'm currently living in Georgia and my wife is Georgian. I know a little, but like to translate when I can to be sure. I get SMS from businesses like my cell service provider for example, that has transliterated Georgian.

Mkhedruli is the most used. The older scripts are still used by the Georgian Orthodox church, but I'm not aware if they use that anywhere online or only on paper documents and iconography.

Many Georgians will use transliterated Georgian in social media posts and instant messaging, but I don't think it's common practice for like a blog, news site or business.

I'm not aware of any Georgian transliterated to Cyrillic, but maybe it happens. I think most of the Cyrillic I see here is Russian. It's hard to be sure, because I know less Russian than Georgian and I don't remember all the Cyrillic characters.

So for convincing the Firefox people, I don't know if you'll find entire websites full of transliterated Georgian, but it is common in social media posts. I think it would be nice to have. However, unless something has changed recently, I don't think Firefox even translates Georgian with Mkhedruli script.