How can I install Firefox on an Android with no service or Google account? This is for a very narrow use case with a phone that is used as a mobile GUI environment only. The phone has no service and is not associated with a Google account, so it can't access Google Play. Is there another way to install Firefox?
You can grab the latest .apk on a PC from Mozilla's FTP server and sideload it, as noted on Mozilla's wiki.

If you're unsure of how to sideload an app, you can find a variety of methods in How can I install an app given only its APK file? (some require internet connectivity, many do not).