Strictly speaking, of course, he didn't use any of those words. He used "naturels." My command of 18th C. French isn't good enough to know exactly what connotations that may have had. I'd say "native" and "indigenous" were slightly different in modern English usage, and we don't use "the naturals" for much of anything.
A French-English dictionary contemporaneous with Vattel defines Les Naturels François as "The French Natives."
I've seen no reason to believe anyone reading the original French version (or English translations available at the time of the drafting of the Constitution) would get "natural-born citizen" from Vattel's Law of Nations.