The wreckage of the plane was in a basin at 2,450 meters (8,000 feet) above sea level surrounded on three sides by peaks, a Reuters witness said.
Operator JU-Air said it was saddened by the news and it had set up a helpline for relatives.
The company's chief executive Kurt Waldmeier told the AFP news agency the plane had undergone a maintenance inspection in July.
The company "still can not explain" the incident, they said, pledging to support the official investigation. Most of the victims were Swiss but they also included a couple and their son from Austria.
Katherine Lam is a breaking and trending news digital producer for Fox News.
The Ju-52 aircraft, made by Germany's Junkers, were made between 1932 and 1952.
Ju-Air offers "adventure flights" for people wanting to experience Switzerland's landscape from vintage planes.
Authorities were informed of the crash at 5 p.m. Saturday, 50 minutes after the aircraft had taken off from Locarno's Magadino airfield. They also expect the investigation into the cause of the crash to be "relatively complex".
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"One can ascertain that the aircraft hit the ground almost vertically at high speed", Daniel Knecht, who heads the aviation division of the Swiss Transportation Safety Investigation Board, said.
Police said that the aircraft had burned and officials haven't yet been able to ascertain the exact number and identity of those who died.
Officials can essentially rule out a collision with another aircraft or an obstacle such as a wire, Knecht said.
JU-Air says it runs a small fleet of four Junker planes, all built in 1939, which are for hire.
The incident was the worst in Switzerland for several years, but just one of three plane crashes in the country in the space of eight days.
It said it was suspending flights until further notice.
Eleven men and nine women were among the people killed on the plane.