The sport of Aeronautics, more accurately described as gliding, has been on the Olympic program only once, as a demonstration sport at the Berlin 1936 Olympic Games. No competition was held, just a demonstration of the sport by 14 pilots from seven countries on August 4th 1936. The sport involved a glider being launched from a bungee.
There was one medal awarded in aeronautics at the 1936 Olympics - Swiss pilot Hermann Schreiber was awarded a gold medal for his flight across the Alps in 1935. Many sources incorrectly join these pieces of information and claim that a gliding competition was held in 1936, with only one participant, Schreiber, who won the gold medal.
The International Olympic Committee approved gliding as an official Olympic Games event, one of the optional group of sports for the 1940 Olympics scheduled in Tokyo. However, those Olympics were canceled because of World War II and gliding never made it on to the Olympic program.
I also found reference to a prize for the most impressive Aeronautic performance, which was awarded twice in the 1920s. I have not been able to find more information on that, though Hermann Schreiber's medal may be one of those.