Season chicken generously with salt and pepper. In a 6-quart Dutch oven or heavy pot, heat oil over medium-high. Cook chicken until brown on all sides, 10 to 12 minutes; transfer to a plate. Add onion, carrot, celery, and garlic to pot; cook until onion softens, 3 minutes. Add potato and collard greens; cook until greens begin to wilt, 3 minutes.
Return chicken to pot; add vegetable juice and Worcestershire. Bring to a boil; reduce to a medium simmer, partially cover, and cook until chicken is separating from the bone and vegetables are tender, about 45 minutes. Season with salt and pepper and stir in vinegar just before serving.