Whisk the lime juice, fish sauce substitute, and sugar together in a large bowl until the sugar dissolves.

Bring 2 quarts of water to a boil in a medium saucepan. Stir in the salt and rice vermicelli, then remove the pan from the heat. Let the noodles sit until they are tender and translucent, about 3 minutes. Drain and rinse with cold water until cooled, then pat them thoroughly dry.

Add the cooled noodles to the bowl with the lime juice mixture. Add the shredded carrot, peanuts, and minced chiles and toss well to coat.

Work with one rice paper wrapper at a time to keep them from sticking. Submerge a wrapper in a shallow dish of warm water until it is just pliable but not completely limp, about 10 to 20 seconds. Lay the wrapper flat on a clean work surface or a damp lint-free towel.

Place one piece of lettuce on the bottom third of the wrapper, leaving a 1-inch border on the sides. Layer a few cucumber matchsticks, a portion of the noodle mixture, and a scattering of basil and cilantro over the lettuce.

Lift the bottom edge of the wrapper and fold it over the filling, tucking it snugly. Fold the left and right sides toward the center, then roll the wrapper away from you into a tight cylinder.

Repeat the process with the remaining wrappers and filling. Serve immediately while the wrappers are still soft and slightly tacky, with the dipping sauce on the side.