My state is good to go:
“Texas
(Signed into Law):
Executive Order GA 35 prohibits state agencies and political subdivisions from adopting or enforcing any order, ordinance, policy, regulation, rule, or similar measure that requires an individual to provide documentation regarding the individual’s COVID-19 vaccination status for any vaccine administered under an emergency use authorization. The Order also prohibits any public or private entity receiving public funds from requiring consumers to provide, as a condition of receiving any service or entering any place, documentation regarding such vaccination status. However, an exception exists to allow (i) nursing homes, (ii) state supported living centers, (iii) assisted living facilities, and (iv) long-term care facilities to require documentation of residents’ vaccination status. The Order took effect on April 5, 2021.
Senate Bill 968 prohibits businesses in the state from requiring “a customer to provide any documentation certifying the customer’s COVID-19 vaccination or post-transmission recovery” to enter, gain access to, or receive any service from such businesses. Businesses that fail to comply with this provision are “not eligible to receive a grant or enter into a contract payable with state funds.” The Bill was signed into law on June 7, 2021 and took effect immediately.”
Neither bill addresses a private corporation requiring its employees to be vaccinated.