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.