In a big bowl, soak the rice-stick vermicelli in hot water to cover until softened and pliable, about 15 minutes; drain in a colander, rinse under cold water, and drain well again.
In a small bowl, whisk the rice vinegar, fish sauce, and sugar together until the sugar dissolves.
Add the vermicelli to a nonreactive salad bowl along with chicken, carrots, cabbage, basil, mint, cilantro, and rice vinegar mixture; toss to combine.
Put a double thickness of paper towel on a work surface, and fill a shallow baking pan with warm water.
Check the rice paper rounds and use only those that have no holes.
Soak 1 round in warm water until pliable, about 30 seconds to 1 minute; carefully transfer to paper towels.
Place ½ cup filling across the bottom third (the part nearest you) of the rice paper; fold the bottom half of the rice paper over the filling and begin rolling it up tightly, stopping at the halfway point.
Fold in the ends and continue rolling.
Transfer the salad roll, seam side down to a plate and cover with dampened paper towels.
Make 9 more rolls in same manner.
To serve, cut the rolls in half crosswise; stand pieces upright; serve with dipping sauce.
The salad rolls can be made 4 hours ahead and chilled, covered with lightly dampened paper towels and then with plastic wrap; let them sit at room temperature for 30 minutes before serving.