Is there a requirement to appoint the candidate who came in second in the most recent election for the office to fill a vacancy?
No. The person appointed by the commissioners court, county judge, or district judge, as appropriate, must meet the qualifications for the office at the time of appointment, including being a registered voter of the territory from which the office is elected. See Tex. Election Code §141.001. For more information on appointments and vacancies, see our publication Pitfalls and Perils of Public Office.