According to this article, this ban is more limited than you think.

“A business in this state may not require a customer to provide any documentation certifying the customer’s COVID-19 vaccination or post-transmission recovery,” the language in the bill begins, only to continue, “A business that fails to comply with this subsection is not eligible to receive a grant or enter into a contract payable with state funds.” Texas pundits assert that very few Texas businesses fall into this category, and thus the overwhelming majority of Texas employers will be permitted to require vaccine passports with little to no negative repercussions.

https://nationalfile.com/deception-abbotts-vaccine-passport-ban-only-prevents-state-contractors-from-requiring-them-hes-just-lying/