Remove the stems from the tomatoes. Put the tomatoes, hot sauce, Worcestershire sauce, lemon juice, and salt into a blender, cover and blend on high speed for 1 1/2 minutes.
Fill an ice cube tray with about 2 cups of the juice and freeze overnight. Pour the remaining juice into a lidded container and reserve, on the counter, overnight.
Just before serving, put 3 frozen cubes into each of 4 Collins glasses and add 1 1/2 ounces of vodka to each glass. Stir the reserved juice, pour 4 ounces into each glass and serve immediately.
Cut each of the tomatoes into 8 pieces and put them into a large glass jar. Add the vodka, stir to combine, and cover. Put in a cool, dark place for at least 5 days and up to 7 days, stirring every day. After 7 days, strain through a fine mesh strainer and discard the solids.
To store, pour the tomato vodka back into its original bottle, and keep in a cool, dark place.