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