Stir water, onion, lemon, garlic, tarragon, seafood seasoning, peppercorns, and bay leaf together in a large pot; bring to a simmer and cook until flavors blend, about 15 minutes.
Whisk ketchup, chili sauce, horseradish, lemon juice, Worcestershire sauce, hot sauce, and salt together in a bowl; refrigerate until chilled, at least 15 minutes.
Bring poaching liquid to a rapid boil. Cook the shrimp in the boiling liquid until they are bright pink on the outside and the meat is no longer transparent in the center, about 5 minutes. Transfer shrimp to a bowl of ice water and immerse in ice water until cold; drain. Arrange cold shrimp on a platter and serve with chilled sauce.
