Whisk the lime juice, vegetarian fish sauce, and sugar in a small bowl until the sugar is completely dissolved.

Bring a large pot of water to a boil, then remove from the heat and stir in the rice vermicelli and salt. Let the noodles sit, stirring occasionally, until they are tender but still have a slight bite, about 10 minutes.

Drain the noodles well, transfer them to a bowl, and toss with a portion of the prepared lime-sauce mixture.

Combine the shredded carrot, chopped peanuts, and minced jalapeño with a portion of the lime-sauce mixture in one bowl, then toss the cucumber matchsticks with the remaining sauce mixture in a separate bowl.

Prepare your workspace by laying a clean, damp kitchen towel on the counter and filling a wide, shallow dish with room-temperature water. Submerge one rice paper wrapper at a time until it is just pliable, about 2 minutes, then lay it flat on the towel.

Scatter basil and cilantro leaves over the center of the wrapper, followed by a layer of cucumber sticks, the carrot mixture, and a portion of noodles, leaving a clear border around the edges.

Fold the bottom of the wrapper up over the filling, tuck in the sides, and roll forward into a tight cylinder. Place the finished roll on a lettuce leaf and keep covered with a second damp towel to prevent the rice paper from hardening.

Repeat the assembly process with the remaining ingredients and serve immediately with peanut dipping sauce, using the lettuce leaves to wrap the exterior of each roll for extra crunch.