Combine bean threads and 2 cups hot water in a bowl; let stand 10 minutes. Drain; cut into 2-inch lengths with scissors.
Add cold water to a large shallow dish to a depth of 1 inch. Cut 4 rice paper sheets in half, leaving remaining 8 sheets whole. Place 1 whole rice paper sheet and 1 half rice paper sheet in dish of water. Let stand 2 minutes or until soft. Remove sheets from water. Place whole rice paper sheet on a flat surface; top with half sheet, lining up edges of both sheets. Place 1/4 cup lettuce over half sheet, leaving a 1/2-inch border around outer edge of half sheet. Arrange 1 tablespoon bean threads, 2 tablespoons bean sprouts, 3 basil leaves, and 4 shrimp over lettuce. Fold sides of rice paper sheets over filling; roll up jelly-roll fashion. Gently press seam to seal; place, seam side down, on a serving platter (cover to keep from drying). Repeat procedure with remaining rice paper sheets, lettuce, bean threads, bean sprouts, basil, and shrimp.
Cut each roll in half crosswise. Combine Lime-Vinegar Sauce and peanuts in a small bowl; serve with rolls.