[...how can we appeal to a moral authority to justify these claims of "Right" and "Wrong" if there is no moral authority?]
You're like a colorblind individual asking how can we know what green is if there's no color authority. There is a general consensus about what green is and therefore what things are green. People are free to go against that grain, but they will be in a small and dismissed minority.
In the case of morality, people with views that go against the grain get to pay fines, go to prison or even be executed. Not because of a moral authority, but because enough people have an innate moral compass to agree on many basic rules. Like don't murder for example. And they don't need to be told this by a dusty tome translated edited censored retranslated re-edited and spoon fed to them before they had the necessary faculties to defend against it. As far as I'm concerned, religion is sociological malware. It's an informational virus that people infect their children with, and most carry and spread that infection their whole life. The whole thing strikes me as terribly immoral, but to each their own.