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/

So does the law

I was concerned it was only going to apply to government institutions like the DMV or courthouses.

If that was the case, all it would take was two grocery stores to require vaccines for entry and they would have the entire state in a stranglehold.

As much as it sucks sometimes, THIS is what government is supposed to be for- to protect the people from the oligarchs.

Wheels still sucks for his heavy-handed Covid response though.

They already can't. It's PHI (Protected Health Information) and it's against Federal HIPAA laws for a business to ask for medial information from you.