Preheat an 8-quart air fryer to 360°F for 3 minutes. Pat chicken dry; rub with oil, salt, paprika, garlic powder, and pepper.
Space breasts skin-side up. Air fry 20 minutes; turn skin-side down and cook 10 minutes.
Mix barbecue sauce and vinegar. Turn chicken skin-side up; brush with half the sauce, reserving rest in a clean bowl.
Increase to 390°F; cook 5 to 10 minutes until coating tacky and thickest breast at least 165°F. Brush with remaining sauce in last 2 minutes.
Rest 5 minutes; debone and cut each breast into 2 portions.