I don't have access to the book right now but I don't recall his citizenship being addressed directly. In the chapter summing up his life, there is some material about how he became politically influential.

My guess is that, sometime when he was building up the olive oil business, he probably contacted whoever fixed such things then and bought the necessary documents. A respectable young businessman would go through the process, so Vito would achieve the same result without the fuss of tests and such.