An analysis of Legend of Korra, the sequel to The Last Airbender, says that due to the technology and advances in medicine and culture from the shows places it in the time period of the 1920s. As Korra was around 16, that places her birth in the early 1900s or mid 1900s, meaning that's when Aang died. Considering the previous avatars lived to be quite old with their own natural deaths, it's very likely that the majority of The Last Airbender's TV run occurred in the 1830s and 40s, at least.
This means that the almighty and infamous cabbage merchant lived in the time period of the contest, which is Feudal Japan/East Asia. According to historians, the majority of people living within the 1100s to 1850s Japan lived by a social hierarchy with Samurais at the top, and Merchants and Artisans towards the bottom, where our Cabbage guy falls.
Throughout the show, the Cabbage Man is victim to the antics of the main characters of the show, always having his cabbages destroyed for some unfortunate reason. His purpose as a character is to typically serve as comedic relief at a tense moment, such as an interrogation, battle scene, or something else that provides stress on Aang and his friends, the main characters.