Korean street food dates back at least 600 years to the Joseon Dynasty circa 1300 AD (and probably much earlier than that). Around that time enterprising Koreans set up food stalls near market areas to sell prepared foods to shoppers, travelers, and others.      

The Korean war (June 1950 - July 1953) left devastation in it's wake. Refugees who had lost nearly eveything were scattered across the country. U. S. troops in large numbers remained in the country. Vendors catered to both the refugees and foreign troops through food carts, or portable stalls that could be moved as needed.
At the end of the Korean War, street vendors offered a cheaper meal option to Koreans attempting to recover from the ravages of bitter warfare. Refugees returning to their war torn homes, villages, and cities relied heavily on street vendors and food stalls for lower cost meals. Additionally, these stalls and food carts earned a nearly livable income from sales of foods to the many U. S. Soldiers and Airmen that were stationed in the country. 
Early contemporary street foods consisted of soups and breads, with jinppang (steamed bread) and hoppang (steamed buns) appearing in the 1960s. Items from US Military commissaries like spam and hot dogs were incorporated In the early 1970s, Kimbap, Tteokbokki, Mandu, Pajeon, Bungeoppang, Fried Chicken, Hodokwaja, Fried Ramen, and more became common street fare.

Soups, stews, and fire roasted vegetables were the first Korean street foods.

Tteokbokki (spicy stir fried rice cake) is one of the most popular street foods. Street vendors and market stalls as well as small hole in the wall shops offer this dish in a multitude of variations.

Bungeoppang
Tteokbokki
Hoddeok
Twiggim
Gyeran Mandu
Grilled Tteokbokki Skewers
Kimbap/Gimbap