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 tomato vodka back into its original bottle and keep in a cool, dark place.