Whisk together first 5 ingredients and 1 tablespoon vinegar in a large bowl. Add cabbage and onion; toss well.
Combine barbecue sauce and remaining 1 tablespoon vinegar in a medium bowl. Add chicken; stir well. (Stir in 1 to 2 teaspoons water if mixture seems dry.) Divide chicken mixture evenly among bun bottoms; top with cabbage mixture, pickle chips, and bun tops.