Morris began riding as a child. In 1952, at the age of fourteen, he won the American Society for the Prevention of Cruelty to Animals (ASPCA) Maclay Horsemanship Finals and the American Horse Show Association (AHSA) Hunt Seat Equitation Medal Final at Madison Square Garden, making him the youngest rider to do so. Morris rode on eight winning Nations Cup teams between 1958 and 1960. He went on to train numerous riders at his Hunterdon stables in New Jersey, who consistently have won in competitions on the East Coast.
Morris has represented the United States in many international equestrian competitions as both a rider and coach. In 1959 his team won the gold medal in the Pan American Games, and he won a team silver medal at the 1960 Rome Olympics.
Students trained by Morris have won medals in the 1984, 1992, 1996, and 2004 Olympic Games. Morris has acted as chef d’equipe for numerous winning teams, including the 2005 champions of the Samsung Super League. He fully assumed the position of chef d’equipe of the United States show jumping team in 2005. He also served in that role for many Nations Cup events.
He coached United States teams to individual and team Silver metals at the 2006 Fédération Équestre Internationale, FEI, World Equestrian Games. In 2008 he coached the team that won the Team Gold Medal at the Olympic Games in Hong Kong as well as the team member who won the Individual Bronze Medal. Morris also serves on the USEF National Jumper Committee and Planning Committee, and he is the president of the United States Show Jumping Hall of Fame.