Bring water to a boil in a saucepan; pour into a mug and brew chai tea for 5 minutes. Remove tea bag and mix brown sugar into the hot tea until dissolved. Allow tea to cool for about 10 minutes.
Mix tea, pineapple juice, rum, vermouth, and vanilla-flavored syrup together in a large punch bowl; stir well. Chill for at least 2 hours before serving.
