Nathan Handwerker and his wife opened their now famous hot dog stand on Coney Island in 1916, but their annual hot dog eating contest did not begin that year—even if legend may tell you otherwise.
The the NY Times has an interview with Mortimer Matz, who worked on promoting the hot dog eating contest in the 1970s with Max Rosey. He tells them Handwerker "was upset that the contestants weren’t paying for the hot dogs and would only let the contest last 12 minutes. We said this was an annual tradition since 1916. In Coney Island pitchman style, we made it up."
Indeed, even though the Times has even cited the 1916 date, there doesn't seem to be any documentation of the contest prior to Matz's involvement in the '70s. A company spokesman told the paper there's no evidence of it on their end, saying they attribute it to legend. However, their website says, "According to archives, the Fourth of July Hot Dog Eating Contest was first held in 1916. The contest has been held each year since then, except in 1941, when it was canceled as a protest to the war in Europe, and in 1971, when it was canceled as a protest to civil unrest and the reign of free love." Perhaps by "archives" they mean this Wikipedia entry.