Generally speaking, private businesses have the right to set their own policies for customers and employees provided they do not discriminate against a protected class of people (e.g., on the basis of race, color, religion, national origin, or disability). A common example of this is a “no shirt, no shoes, no service” rule you may see at some businesses.
In providing or obtaining services, every person (including individuals, businesses, and other legal entities) is strongly encouraged to use good-faith efforts and available resources to follow the Texas Department of State Health Services (DSHS) health recommendations, found at www.dshs.texas.gov/coronavirus
Previous COVID-19-related executive orders governing mask usage have been rescinded and replaced with GA-38. Prior to GA-38, Governor Abbott's Executive Order No. GA-34 [PDF] stated that “nothing in the order precluded businesses and other establishments from requiring employees or customers to follow additional hygiene measures, including the wearing of a face covering.” However, this language does not appear in GA-38.
If you have a disability that affects your ability to wear a face mask, you may consider asking the business for an accommodation. See our Legal FAQ Do I have to wear a mask if I have a disability? for more on this subject.