Generally, septic systems, also known as on-site sewage facilities, are regulated under a portion of the Texas Health and Safety Code, which you can find here and the Water Code, here
With a few exceptions, a permit is required to install a septic system in Texas. The Legislature, in the Health and Safety Code, designates the Texas Commission on Environmental Quality (TCEQ) as the official agent to investigate and decide whether to approve a permit request.
The commission is also tasked with overseeing whether the septic tanks continue to function properly and the Commission can require the property owner make repairs when needed.
However, the law allows the Commission to designate a local authority, when there is one, to handle the above tasks. In some cases, this authority could be the local or county health department or a river authority.
For more information about the permit process as well as a link to find local permitting authorities, go here. Finally, one can read about minimum state standards in the Texas Administrative Code, here.