In a large pitcher, combine the vegetable cocktail, celery seed, pepper, citrus juices, 2 tablespoons grated horseradish, Worcestershire sauce, and hot pepper sauce and stir well to combine. Cover and refrigerate for 1 hour.
Fill 4 glasses with ice. Add 2 ounces vodka to each and fill three-quarters full with the vegetable cocktail mix. Sprinkle with the remaining horseradish, place 1 celery stalk in each glass, and squeeze a lime wedge over each glass before dropping the wedge into the glass. Float a teaspoon of balsamic vinegar on the top of each drink and serve.