Whisk rice vinegar, 2 Tablespoons barbecue sauce, and garlic together in a large bowl.
Place chicken, skin-side down, in the bottom of the marinade bowl, cover the bowl with plastic wrap, and refrigerate for an hour.
Preheat an outdoor grill to medium-high heat.
Take a few paper towels and fold them several times to make a square. Blot a small amount of oil on the paper towel and carefully wipe the hot grates of the grill to make a nonstick surface.
In a small bowl, mix together the salt, pepper, paprika, onion powder and cayenne pepper and set aside.
Remove the chicken from the marinade, pat the chicken dry with paper towels, and discard the marinade.
Place chicken pieces, skin side up, on a plate and on all sides, season with the seasoning mixture.
Cook the chicken, skin-side down, on the preheated grill for 3-4 minutes, or until grill marks appear.
Turn the heat down to medium heat. Turn chicken over, close the lid of the grill, and cook, basting with barbecue sauce every 5 minutes, until no longer pink at the bone and the juices run clear, about 35-40 minutes.
Check for doneness after 30-35 minutes with an instant read thermometer inserted into the thickest part of the meat, near the bone should read 160º for breasts and 170º for thighs.
Serve with extra sauce.