Texas has no ESA statute of its own
Here's something many Texans don't expect: no Texas statute mentions emotional support animals by name. The Texas Fair Housing Act, found in Property Code Chapter 301, uses general language instead: under §301.025(c)(2), saying no to a reasonable adjustment of house rules or practices is treated as disability discrimination if a resident with a disability may need that adjustment to enjoy the home as fully as anyone else.
That means an ESA request in Texas is simply a reasonable-accommodation request, weighed under that state provision alongside the federal FHA. At the state level, complaints go to the civil rights arm of the Texas Workforce Commission (TWC).