No, he was a "Ritter" (Knight) and thus entitled to the 'von'. When the socialists took over Austria after World War I, they abolished the titles, but von Trapp was entitled to Italian citizenship, and Italy retained titles.
So, in Austria, the country that awarded the title, he would have been called Trapp; in Italy, whose ships he sank, he would have been called von Trapp.
Here you go: https://www.archives.gov/files/publications/prologue/2005/winter/images/manifest-pg1-l.jpg
Herr von Trapp entered the US as an Italian citizen with an anchor baby born in Philadelphia on an earlier visit.