A member of the Mohawk nation, Alwyn Morris took up canoeing after realizing that his size made him better suited for this sport than lacrosse, which had been his first passion. By 1977 he was winning junior national championships and originally intended to join with Hugh Fisher to compete in K2 events in 1980, but this partnership was delayed after Fisher sustained a wrist injury that took him out of the 1981 season. In the interim Morris competed with Denis Barré and the two were successful enough that the Fisher-Morris partnership was nearly abandoned. When Barré suffered an injury of his own, however, Fisher was brought back into the fold and the duo won silver in the K2-1000 at the 1982 World Championships. The following year they earned bronze in the K2-500 at the tournament and were then selected to represent Canada at the 1984 Summer Olympics, where they captured gold in the K2-1000 and bronze in the K2-500. While standing on the podium, Morris raised the feather of an eagle as a tribute to his ancestry.
Following the Games Morris was appointed to serve as a Canadian youth ambassador and was later selected for the 1988 Summer Olympics, where Fisher and Morris were eliminated in the semifinals of the K2-500. They attempted to qualify for the 1992 Games, but failed to do so and retired from active competition. As of 2013 Morris is the CEO of the Morris Mohawk Gaming Group and, in addition to being named a Member of the Order of Canada in 1985, he was inducted into Canada’s Sports Hall of Fame in 2000.