The von Trapps had a performance booking in Norway in September 1939. As they were planning their departure from home in Salzburg, war broke out and the father Georg Ludwig Ritter von Trapp decided that they would not return. As soon as they got to Norway, they applied to the USA Embassy for entry to USA. They had performed in America the previous year and were known to the Embassy staff.
Austria was annexed [Anschluss - taken over by a bloodless invasion] by Nazi Germany in March 1938. As a naval officer, Georg von Trapp would have been commissioned into the Nazi Kriegsmarine. He applied for exemption on the basis of his age of 58, and was excused service. Remember that Austria has no coastline, so German officers regarded Austrian naval officers as a joke.
Georg vonTrapp was not strictly Austrian. He was born in Dalmatia, then part of the Austro-Hungarian Empire, now in Croatia. He kept to himself his views about Anschluss for the safety of his family, but later it became clear that he despised the Nazi minority which had taken over his country.
Gradually after 1938, Austria fell into the Nazi model of antisemitism, secret police, prison camps, and neighbours "informing" on each other about real or imaginary breaches of Nazi laws and regulations. Georg von Trapp planned from that time to take his family abroad if conditions got too bad.
The family had been invited to perform in New York City at The Town Hall on 10 December 1938, and Georg von Trapp had an exploratory conversation with an American lawyer about obtaining right to remain. The lawyer was typical of most Americans at the time; they regarded the situation in Europe as nothing to do with USA, and just another foreign squabble. Von Trapp was told not to bother enquiring further.
The family left Norway for USA on 22 September 1939 and settled in Vermont. The two older sons Rupert and Werner became citizens by serving during World War II. Maria von Trapp applied for USA citizenship in 1944. There is a copy of her application in the public domain; see source below. Georg von Trapp never applied for citizenship.
Georg von Trapp died in Vermont, from lung cancer on 30 May 1947, aged 67. Maria von Trapp died of heart failure on 28 March 1987, in Morrisville, Vermont, aged 82.