Combine everything but the chicken into a food processor and blend into a smooth paste.
Place chicken in a large ziplock bag, add marinade and shake around until all drumsticks are evenly coated with the marinade. Place in fridge and let marinate for 30 minutes-1 hour. Bring to room temperature before cooking.
Once grill is heated, place drumsticks onto grill rack for about 20 minutes total, turning every 5-6 minutes, until no longer pink and cooked through.