I recently attended a wedding in Pennsylvania where the bride and groom essentially married themselves, using a special type of marriage license which permits this. They had a friend “officiate.” Since he was not legally required, he needed no legal authority. I suspect New York has some similar procedure, so Clinton would not need any authority to preside.
I had a boy friend in college who used to try that line on me. (Just you and I under the stars with God as our witness. It will be just the same as if we were married. We don't need a church.) It never worked. I thought that scam went out in the '50s. LOL.