Make the Tomato Dipping Sauce:
In a medium bowl, combine all the ingredients. Cover with plastic wrap and refrigerate until ready to serve.
Make the Soft-Shell Crab Tempura:
In a large saucepan, heat 3 inches of oil over medium-high heat to 375°F.
In a medium bowl, whisk the ice water, egg yolk, chives, and salt until frothy. Add the flour and whisk just until blended and small lumps remain. Dip the crabs in the batter, then carefully add to the oil, 2 at a time, and fry for about 5 minutes, until cooked through and golden brown. Using a slotted spoon, transfer to paper towels to drain.
Place each crab on a serving plate and spoon some tomato dipping sauce onto each plate. Garnish with lemon wedges and serve immediately.
Note: Ask your fish market to clean your crabs if possible. If you're cleaning them yourself, fold back the flaps and remove the gills. Cut off the side aprons and head with kitchen shears. Pull the pointed flap on the underside down and twist it off. Remove the tomalley (liver) and discard.