This is nonsense. They were not citing a book. They were citing a term. The same term that an author used to title his book.
Vattel's book is not the Constitution. The Constitution is not a blanket endorsement of everything Vattel ever wrote. This is not rational thinking.
In English common law, which is not what Vattel was talking about but is what applied in the colonies, a "natural born" subject is simply someone born within the country.
It was common practice at the time the Constitution was written to capitalize important traditional phrases such as Common Law, the Law of Nations, or Natural Law.
Or, for example, Original Sin, or Constitution. This was to indicate that you were not talking about any old sin but Original Sin, and not any old constitution but THE Constitution.