Put the vanilla yogurt, orange juice, and 2 cups of the orange soda in the blender container. Cover blender with lid. Blend on high speed until combined. Turn off blender.
Pour the mixture into the ice cube trays. Cover the ice cube trays with plastic wrap. Put the ice cube trays in the freezer. Freeze for 6 hours or overnight.
Just before serving, remove the frozen cubes from 1 of the ice cube trays. Put the cubes in the blender container. Add 1 cup of the orange soda. Cover and blend on high speed until slushy. Turn off the blender. Pour into 6 glasses. Repeat with remaining frozen cubes and orange soda. Makes 12 (1/2-cup) servings.