Vincent Viola, a West Point graduate who rose from trading gasoline futures on the floor of the New York Mercantile Exchange to build one of the world’s largest high-speed trading firms, was tapped by President-elect Donald Trump as the next secretary of the Army.

Mr. Viola, who goes by “Vinnie,” is well connected in both financial markets and national security circles. He is the billionaire founder and executive chairman of Virtu Financial Inc., a trading firm active on more than 200 exchanges world-wide. A U.S. Army veteran,...