Soak vermicelli in 2 cups warm water for about 15 minutes. Drain and set aside.
Dip sheet of rice paper in warm water (110 °F) and transfer to work surface covered with a damp kitchen towel. Wait about 30 seconds or until wrapper is pliable. Place a lettuce leaf on the bottom two-thirds of the rice paper, leaving a 2-inch border of paper on the bottom. Place 2 tablespoons vermicelli, 1 tablespoon shredded carrots, 2 slices of mango, 1 tablespoon each of basil and mint, and 2 tablespoons bean sprouts on top of the lettuce. Fold up the bottom 2-inch border of rice paper over the filling. Fold upward again to enclose the filling. Fold in the right, then the left edges of the wrapper. Continue folding until a tight cylinder is formed.