Whisk olive oil, cider vinegar, paprika, basil, celery salt, mustard powder, cinnamon, and red pepper flakes together in a bowl; pour into a resealable plastic bag.
Add drumsticks, coat with the marinade, squeeze excess air from bag, and seal. Marinate in the refrigerator, 8 hours to overnight.
Preheat grill for medium heat and lightly oil the grate.
Remove drumsticks from marinade and shake to remove excess moisture.
Pour marinade into a small saucepan and bring to a boil. Reduce heat to medium-low and simmer for 10 minutes; set aside.
Cook drumsticks on preheated grill, basting frequently with heated marinade and turning 1/4 turn every 12 minutes, until no longer pink at the bone and the juices run clear, about 48 minutes total. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).
