Light a gas grill. Wrap the hickory wood chips in heavy-duty foil and pierce the packet all over with a skewer. Add the foil packet to the grill. Cover and let the wood chips begin to smoke, about 5 minutes.
Brush the chicken with oil and season generously with salt and pepper. Grill the chicken, covered, over moderately high heat, turning and shifting the pieces occasionally, until lightly charred and cooked through, 35 to 40 minutes. Transfer to a platter.
In a medium bowl, whisk the mayonnaise with the vinegar, apple juice, horseradish, lemon juice and cayenne and season with 1 teaspoon of black pepper and 1/2 teaspoon of salt. Transfer the barbecue sauce to a pitcher and serve with the chicken.