Make the dipping sauce: in a small bowl whisk together sour cream, mayonnaise, miso paste, lemon juice, minced garlic, and chopped tarragon until smooth; season with pepper and refrigerate until ready to serve.
Cook the vermicelli according to package instructions, drain in a colander, rinse with cool water, and transfer to a large bowl.
Prepare the vegetables and herbs: ribbon or julienne the carrot, thinly slice the avocado and cucumber, chop the mint, tear the basil, and pick the chives.
Fill a large shallow bowl with cool water. Submerge one rice paper sheet briefly until pliable, then lay it flat on your work surface.
Assemble the roll: place a portion of vermicelli, lettuce, shredded chicken, carrot, cucumber, avocado, and herbs across the bottom third of the rice paper.
Fold the sides of the rice paper in over the filling, then fold the top down to lightly touch the filling and roll away from you, tucking the filling in as you go to make a firm roll without tearing the paper.
Repeat with remaining wrappers and filling. Serve the rolls whole or halved with the miso-tarragon dipping sauce.