Vattel was far from obscure to the well educated men who wrote the Constitution.
Delegates to the First and Second Continental Congress, which produced the Declaration of Independence, often consulted The Law of Nations, as a reference for their discussions. One important reason why the delegates chose to meet in Carpenters Hall, was that the building also housed the Library Company of Philadelphia. The librarian reported that Vattel was one of the main sources consulted by the delegates during the First Continental Congress, which met from Sept. 5 to Oct. 26, 1774.
Charles W.F. Dumas, an ardent supporter of the American cause, printed an edition of The Law of Nations in 1774, with his own notes illustrating how the book applied to the American situation. In 1770, Dumas had met Franklin in Holland, and was one of Franklin's key collaborators in his European diplomacy. He sent three copies to Franklin, instructing him to send one to Harvard University, and to put one in the Philadelphia library. Franklin sent Dumas a letter, Dec. 9, 1775, thanking him for the gift. Franklin stated, "I am much obliged by the kind present you have made us of your edition of Vattel. It came to us in good season, when the circumstances of a rising state make it necessary frequently to consult the law of nations. Accordingly, that copy which I kept, has been continually in the hands of the members of our congress, now sitting ... ." And that was just the most recent English translation. Franklin, and many of the other founders, would not have needed it, since they could read the original French.
That doesn't explain why the founding fathers would use a definition without explanation from a theoretical work casting aside a definition of citizenship familiar to all.