In a medium pot, bring 1 1/3 cups water to a boil. Add the rice, cover and turn the heat to low. Cook until all the water has been absorbed and rice is tender, about 15 minutes. Let cool until warm.
To the pot with the rice, stir in the milk, sugar and salt. Bring to a simmer, turn the heat to low and cook until the mixture is thickened, about 10 minutes. Remove from the heat, stir in the chocolate chips and vanilla and let rest for 5 minutes. Serve warm or cover and cool in refrigerate for at least 1 hour.