You say they have quoted Vattel, OK, did they quote him on this specific subject saying that is where they got the meaning. I would presume there were a number of books at the convention. Madison asked Jefferson to send him many books from Paris. The answer is yes. There are some - not many, but some early quotations and references - indicating fairly conclusively that the Vattel definition is precisely what the framers had in mind. These can be researched on the web. The Federalist Society is a good place to start.
A search of The Federalist Society Web site for Vattel produced
these results.
Natural born citizen produced
these results.
Nothing quoting the Framers advocating the Vattel view of natural born citizenship.
A google search led me to birther.org of all places, which compares Vattel to Blackstone. I find it less than convincing, but then reasonable minds can disagree. The two quotes I see trying to directly link this are the Jay quote, which can be attributed to the idea that went around that the US should import a king. The other is the virginia citizenship statute written by Jefferson. It may be a technicality, but I don't think he would qualify as a Framer as he was in France at the time of the convention.
The rest of the article seems to say if Vattel influenced them on this, then he must have influenced them on the other thing. I don't find that convincing. Especially since the natural born citizen phrase is never discussed and it would seem then to default to what would be the understanding of the average citizen which would be English common law or as close as reasonably possible under the different form of government.