In a small saucepan, bring the apple cider to a boil. Remove from heat and add the chai tea bags. Steep for 15 minutes.
Discard the tea bags. Stir in the honey. Bring the cider and honey to a boil until slightly reduced, about 5 minutes. Remove from heat and stir in the vanilla. Let cool.
In a large pitcher or punch bowl, stir together the caramel vodka, heavy cream, and apple chai syrup. Chill. If desired, shake individual servings in a cocktail shaker with an egg white to create the froth.