In a large bowl, whisk together ketchup, soy sauce, garlic, vinegar, brown sugar, lemon zest and juice, ginger, and salt; transfer half of mixture to a small saucepan.
Whisk oil into remaining ketchup mixture in large bowl; stir in chicken until well coated. Cover and refrigerate for 1 hour.
Spray grill rack with nonflammable cooking spray. Preheat grill to medium-high heat (350° to 400°).
Grill chicken, covered with grill lid, over indirect heat, turning frequently, until an instant-read thermometer inserted in thickest portion registers 165°, 30 to 40 minutes. Remove from grill, and let stand for 10 minutes.
Meanwhile, bring reserved sauce in saucepan to a boil over medium heat, stirring frequently; brush onto chicken. Serve remaining sauce with chicken.
Recipe by Paula Deen Magazine at https://pauladeenmagazine.com/grilled-sweet-sour-drumsticks/