Stir the yogurt, sugar, and vanilla extract together until the sugar has dissolved. Cover. Refrigerate 1 hour.
Pour the chilled mixture into an ice cream maker and freeze according to manufacturer’s directions until it reaches “soft-serve” consistency. Transfer to a one- or two-quart plastic container with a lid; cover surface with plastic wrap and seal. For best results, frozen yogurt should ripen in the freezer for at least 2 hours or overnight.