As a bobsledder Stöckli finished fourth in the four-man event at the 1952 Winter Olympics. Next year he won a gold medal in the two-man event at the 1953 FIBT World Championships in Garmisch-Partenkirchen. A week later in the four-man event, Stöckli would escape with minor injuries by landing on top of an American jeep after being thrown from the sled that took the life of his fellow Swiss Felix Endrich. Stöckli retired soon after that accident.