As far as what version of Vattel the framers would have access to, here it is:
Translating the French text: "les naturels, ou indigenes, sont ceux qui sont nes dans le pays, de parents citoyens"
To English, gives this: "the natural, or indigenous, are those born in the country, parents who are citizens"
Forgive me, for my French is not all that great, as I am only laughed at when I try to order a meal in a Paris bistro, but where in the translation is the term "Natural Born Citizen?"
And if the Framers based the Constitution on the French writings of Vattel, how is it they invented the term "Natural Born Citizen?"