Here’s an interesting list of the authors cited by the Founders, looking at all their writings as a whole.
http://www.constitution.org/primarysources/influences.html
Vattel is #30 of 37 at .50%. This compares to Montesquieu at #2 (8.3%). Most notably, Blackstone, who held views on “natural born-ness” quite different from those of Vattel, came in at #3 (7.9%).
Vattel actually tied for last place with 12 other writers.
To me this is a very good indication Vattel was a quite minor influence on the Founders. He was not unknown, but was vastly outweighed in importance by Blackstone and others.
-PJ