Generally, private businesses can set their own mask policies for customers and employees provided they do not discriminate against a protected class of people.
Section 81B.002 of the Texas Health & Safety Code prevents state and local government entities from issuing mask mandates. These restrictions were in Senate Bill 29, which added Chapter 81B to the Texas Health & Safety Code. The new law became effective September 1st.
However, this law does not apply to private entities. Businesses can set their own rules and policies — like a "no shirt, no shoes, no service" rules you may see in a store. The policies cannot discriminate against a protected class of people (e.g., on the basis of race, color, religion, national origin, or disability).
People with certain disabilities may be entitled to reasonable accommodations to face mask policies under the Americans with Disabilities Act (ADA) and other disability laws.
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.