As stated by the Texas Alcohol Beverage Commission, individuals cannot legally buy liquor in Texas on Sundays. This is part of the state's "blue laws," which are a set of laws that prohibit the sale of items due to Sunday being for rest and church. The prohibition of liquor sales also extends to Thanksgiving, Christmas and New Year's Day.
While you cannot buy liquor at stores in Texas on Sunday, you can order alcoholic drinks at restaurants and bars after noon. You are also able to buy wine and beer at stores that day (also after noon), just not liquor. The TABC also notes that alcoholic beverages are allowed at major sports venues and festivals or concerts starting at 10 a.m. on Sundays.