At his local sports club Armand Blanchonnet trained in running, swimming, and cycling, specialising in the latter. In 1924 Blanchonnet competed in the French national road race championships for amateurs and finished second behind René Hamel. The result earned both riders places at the Paris Olympics, with Blanchonnet going on to dominate in the road race, winning by almost 10 minutes, ahead of Rik Hoevenaers of Belgium in second and Hamel in third. The two other French riders, Georges Wambst and André Leducq, finished eighth and ninth, respectively, with the home riders also winning gold in the combined team event. A few weeks after the Olympics, Blanchonnet capped off the year by winning bronze in the amateur road race at the World Championships.
Despite his initial success Blanchonnet only went on to win a small number of races after the Olympics. He focused on six-day racing, both in France and across the Atlantic, finishing third in an event in Detroit in 1928. Back home he won six-day titles in Paris in 1930 and in Marseille in 1932. Sandwiched between these two wins Blanchonnet also became the French road race champion in 1931. He retired from the sport a few years later and worked in the insurance sector.