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 (or the body) 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 his or her successor would have to have qualified for office (to cure the holdover issue). See AG Op. C-452 and Short Answers to Common Questions: Vacancy.