Mix 3 tablespoons vinegar, sugar, salt, and pepper together in a bowl. Mix in cucumbers.
Heat oil in a skillet over medium heat. Add cabbage and white parts of scallions to the skillet; cook, stirring occasionally, until softened, 3 to 4 minutes. Transfer to a bowl and toss with salt and pepper and remaining vinegar.
Season chicken with salt and pepper. Cook in the same skillet until juices run clear, 4 to 7 minutes. Stir in sauce; cook until reduced, about 2 minutes more.
Mix sour cream, water, and sriracha sauce together. Spread some of the mixture over each tortilla. Top with bulgogi chicken, cucumbers, cabbage, and green parts of scallions.