Stir the beer, brown sugar, salt, paprika, black pepper, cayenne pepper and teriyaki marinade together in a bowl.
Rub the drumsticks thoroughly with the vegetable oil. Poke several holes in the flesh of the chicken legs with a fork; arrange in a single layer in a glass dish. Pour the marinade over the drumsticks; marinate in refrigerator for 1 hour.
Preheat barbecue for medium heat and lightly oil the grate.
Remove the chicken from the marinade and shake off any excess liquid. Discard remaining marinade.
Cook the drumsticks on the preheated barbecue until no longer pink at the bone and the juices run clear, turning one-quarter turn every 5 minutes, 20 to 25 minutes total.