May a sitting member of the commissioners court be appointed to fill a vacancy?
No. Members of the commissioners court are not eligible to appoint themselves ("self-appointing") to county judge or any other office to which they have authority to make appointments. To be eligible to be appointed by the commissioners court to fill a vacant office, a county commissioner would have to resign and the commissioner's successor would have to qualify for office (to cure the holdover issue) before being appointed. See Texas Attorney General Opinion No. WC-0452 (1965) and Pitfalls & Perils of Public Office: Vacancies.