Put oven rack in middle position and preheat oven to 350°F.
Whisk together mochiko, sugar, baking powder and salt in a large bowl. Whisk together coconut milk, eggs, butter, and vanilla in another bowl. Add coconut mixture to flour mixture, whisking until batter is combined.
Pour batter into an ungreased 13- by 9-inch baking pan, smoothing top, and bake until top is golden and cake begins to pull away from sides of pan, about 1 1/2 hours. Cool cake completely in pan on a rack, about 2 hours. Cut mochi into 24 squares before serving.