Place radish lengthwise on a flat work surface. Trim ends and slice off the top third so it is flat. Turn and cut into rectangular, 1/16-thick slices using a mandoline slicer.
Mix 2 tablespoons vinegar, 2 tablespoons sugar, and salt together in a small bowl until sugar dissolves. Place the radish rectangles in bowl, turning to coat. Marinate until soft enough to roll, about 1 hour.
Trim radish sprouts to be 3 inches in length.
Set aside any cucumber matchsticks without skin for another use.
Squeeze excess liquid from the radish rectangles. Place some radish sprouts and a slice each of cucumber, yellow bell pepper, and carrot on 1 rectangle; roll up. Repeat with remaining sprouts, cucumber, yellow bell pepper, carrot, and radish rectangles.
Mix remaining 1 tablespoon vinegar, remaining 1 tablespoon sugar, and hot pepper paste to make a dipping sauce. Serve sauce with radish wraps.
