Sayle Unit, Texas Department of Criminal Justice, Institutional Division.
General Information Guide for Families of Offenders, page 5:
Search of Visitors and Vehicles: Visitors and their vehicles can be searched before entering the unit. Any visitor refusing the search procedure will have to leave the property. All visitors will be screened by a metal detector prior to visiting an offender. Cash (except coins), alcoholic beverages, tobacco, cell phones, drugs, firearms, escape implements, or dangerous weapons are strictly prohibited by statute and TDCJ rules and policies.
Note that the rules for entrance for visiting are not the same as the rules for the parking lot. TDC and/or Sayle Unit might or might not deny entrance to the parking area based on a handgun in the car; if they do, that exceeds their statutory authority, but you're not going to win the fight at the gate.
I suggest calling the unit, and phrasing the question like this: "When I come for a visit, how do I secure my handgun before I enter the grounds? Do I need a lock box, or is locking it in the trunk good enough?"