Lightly brush grill rack with vegetable oil. Prepare grill with coals stacked to one side or heat gas grill to medium-low heat.
Sauté onion and garlic in oil in skillet until softened, 5 minutes. Mix onion, garlic, vinegar, ketchup, pickle relish, mustard, Worcestershire, molasses and hot sauce in a small bowl. Season chicken with salt and pepper.
Grill chicken, uncovered, directly over coals for 5 minutes, until browned. Turn over; spoon sauce on chicken. Grill 5 minutes. Turn over; spoon on more sauce. Move to cooler side of grate. Cover grill. Grill 10 minutes. Spoon on sauce. Turn every 5 to 10 minutes, until internal temperature registers 165 degrees F, about 50 minutes more. Do not spoon on sauce during last 10 minutes.