Whisk together buttermilk, eggs, salt, and pepper and pour into a 3-quart shallow dish. Add crabs and soak, covered and chilled, for 30 minutes to 1 hour.
Heat enough vegetable oil to measure 1-inch deep in a wide shallow skillet to very hot.
Whisk together flour and cornmeal and transfer to a large sealable plastic bag. Lift 1 crab out of buttermilk, letting excess drip off, and shake in bag to coat with flour. Knock off excess flour and transfer to a tray. Repeat with remaining crabs, arranging them in 1 layer as coated. Shallow-fry the crabs, 4 at a time, turning over halfway through frying (watch out for popping or use a frying screen), until golden brown, 3 to 5 minutes. Reheat oil to very hot between batches. Drain crabs on paper towels and serve with tartar sauce.
Stir together all sauce ingredients. Refrigerate until ready to use.
If you can't find self-rising cake flour, you can use 1 cup all-purpose or cake flour mixed with 2 teaspoons baking powder and 1/4 teaspoon salt.