In a large bowl, whisk together the olive oil, apple cider vinegar, Worcestershire sauce, soy sauce, brown sugar, Weber Smoky Mesquite Seasoning, garlic powder, onion powder, smoked paprika, black pepper, and salt until well combined.
Add the chicken legs to the marinade, ensuring they are fully coated. Cover the bowl with plastic wrap or transfer the chicken legs and marinade to a resealable plastic bag. Refrigerate for at least 2 hours, or preferably overnight for maximum flavor.
Preheat your grill to medium-high heat (about 375°F).
Remove the chicken legs from the marinade and let any excess drip off. Discard the marinade.
Place the chicken legs on the grill and cook for about 30-35 minutes, turning occasionally, until the internal temperature reaches 165°F and the skin is crispy and caramelized.
Remove the chicken from the grill and let it rest for 5 minutes before serving.