The writings of Founders and records will show that Franklin and Washington and probably others were well aware of Vattel and it was Franklin who had copies of Vattel’s books who suggested that the drafted Constitution be revisited to incorporate the the Founder’s intent to have a ‘natural born citizen’ as a requirement for POTUSA. Cruz being born and resident in Canada for four years of a USA citizen mother and a Cuban father does not meet the POTUSA requirements as far as I am concerned. It can be argued that the USA shorts Itself from very able and really Constitutional people by the Founders use of words but I believe the Founders were men of wisdom beyond their times and did their best to lay out a document ‘to form a more perfect union’ for generations to come. There are and will be men and women to carry the Founder’s intentions.
Of course the Founders were well aware of Vattel. They were well aware of a lot of people. They studied literally dozens of philosophers and political writers.
Vattel was a writer on international law, and was influential in that sphere. There's nothing to suggest the Founders paid the slightest attention to him when it came to citizenship, or a lot of other things, for that matter. He believed the government should control religion and restrict freedom of speech, and that only the upper class should be able to keep and bear arms. They clearly rejected him on all those points, and there is nothing to even faintly suggest they followed his ideas on citizenship.
And no, there is nothing to suggest that Ben Franklin was the person who pushed the natural born citizen qualification. The person who did that was John Jay, and there’s nothing to suggest he did anything other than write a letter to Washington.