In a bowl, whisk together apple cider vinegar, oil, worcestershire sauce, hot sauce and salt. Remove 1/2 cup of the marinade and place into an airtight container and refrigerate, this is for basting ater. Add drumsticks to the bowl and turn to coat. Cover and marinate in the refrigerator for 4 hours. You can also do this in a ziplock bag.
Remove drumsticks from marinade. Preheat grill over medium heat. Once hot, add drumsticks to grill and cook for about 30 minutes, flipping and basting with reserved marinade every 5 minutes, until the chicken reaches 165 degrees F.