Here's why I said you should consult a lawyer. Your employer cannot prohibit you from having a weapon and/or ammunition in your car if you are legally authorized to do so.
SUBCHAPTER G. RESTRICTIONS ON PROHIBITING EMPLOYEE
TRANSPORTATION OR STORAGE OF CERTAIN FIREARMS OR AMMUNITION
Sec. 52.061. RESTRICTION ON PROHIBITING EMPLOYEE ACCESS TO
OR STORAGE OF FIREARM OR AMMUNITION. A public or private employer
may not prohibit an employee who holds a license to carry a
concealed handgun under Subchapter H, Chapter 411, Government Code,
who otherwise lawfully possesses a firearm, or who lawfully
possesses ammunition from transporting or storing a firearm or
ammunition the employee is authorized by law to possess in a locked,
privately owned motor vehicle in a parking lot, parking garage, or
other parking area the employer provides for employees.
However, the law does NOT say that the employer can't ask you about it. It merely says your employer does not have a duty to do so.
(c) For purposes of this section, a public or private
employer, or the employer's principal, officer, director,
employee, or agent, does not have a duty:
(1) to patrol, inspect, or secure:
(A) any parking lot, parking garage, or other
parking area the employer provides for employees; or
(B) any privately owned motor vehicle located in
a parking lot, parking garage, or other parking area described by
Paragraph (A); or
(2) to investigate, confirm, or determine an
employee's compliance with laws related to the ownership or
possession of a firearm or ammunition or the transportation and
storage of a firearm or ammunition.
So, can your employer legally ask you to prove that you are authorized to have weapons and/or ammunition in your possession in your car? I don't know. A lawyer is required to know.