Whisk 2 cups plain whole-milk yogurt, 1 1/2 cups simple syrup, 1 teaspoon finely grated lime zest, 3 tablespoons fresh lime juice, and 1/2 cup water in a medium bowl; scrape in seeds from 1/2 vanilla bean. Process in an ice cream maker according to manufacturer's instructions. Transfer to an airtight container; cover and freeze until firm, at least 2 hours.