Place strawberries, sugar, vanilla extract, and half-and-half in a blender; puree until smooth. Add yogurt (see Cook’s Note) and pulse until combined.
Pour the mixture into an ice cream maker and freeze according to manufacturer’s instructions. Transfer frozen yogurt to a lidded plastic container. Cover surface with plastic wrap and seal. For best results, let it ripen in the freezer for at least 2 hours or overnight before serving.