No, it is a fairly small sub - regional, controlled by its founding family. Because of their service and pricing, they have managed to hold off the behemoths (to use your words). Territory expansion has tended to come through their niche Central Market stores. (They have only a few hundred stores in Texas, and some operations in Mexico.) There are still Butts running Harry E. Butts Grocery stores. So it IS a mom & pop operation.
You must be confusing them with some other chain.
HEB Grocery Company, LP (stylized as H-E-B), also known as H-E-B Grocery Stores, is a privately held supermarket chain based in San Antonio, Texas, with more than 350 stores throughout the U.S. state of Texas, as well as in northern Mexico. The company also operates Central Market, an upscale organic and fine foods retailer.
As of 2012, the company has a total revenue surpassing $18 billion USD (2012). H-E-B ranked No. 12 on Forbes’ 2012 list of “America’s Largest Private Companies.” H-E-B was named Retailer of the Year in 2010 by Progressive Grocer. Supermarket News ranked H-E-B No. 13 in the 2008 “Top 75 North American Food Retailers.” Based on 2010 revenues, H-E-B is the twenty-fifth largest retailer in the United States.
It donates 5 percent of pre-tax profits to charity. It has been labeled a religious company; until 1976 it was closed on Sundays and did not sell alcohol.