Below are the cities that the US Postal Service accepts for the ZIP code 78076. This code is specific to the city of Zapata, Texas, and its surrounding areas. If you are mailing a letter or package to this area, be sure to use this ZIP code to ensure that your mail is properly delivered. The city listed in a ZIP code may not be the preferred city for that area. The city associated with the ZIP code 78076 is typically the name of the main post office for that area. When sending your package or letter, be sure to include the preferred or acceptable city. Using any city not on the list of acceptable cities may result in delays.