Tex. Election Code § 145.036

(a) Except as provided by Subsections (b) and (b-1), if a candidate's name is to be omitted from the ballot under Section 145.035, the political party's state, district, county, or precinct executive committee, as appropriate for the particular office, may nominate a replacement candidate to fill the vacancy in the nomination.

(b) An executive committee may make a replacement nomination following a withdrawal only if:

(1) the candidate:

(A) withdraws because of a catastrophic illness that was diagnosed after the first day after the date of the regular filing deadline for the general primary election and the illness would permanently and continuously incapacitate the candidate and prevent the candidate from performing the duties of the office sought; and

(B) files with the withdrawal request a certificate describing the illness and signed by at least two licensed physicians;

(2) no political party that held primary elections has a nominee for the office sought by the withdrawing candidate as of the time of the withdrawal; or

(3) the candidate has been elected or appointed to fill a vacancy in another elective office or has become the nominee for another office.

(b-1) An executive committee may make a replacement nomination following an administrative declaration of ineligibility under Section 145.003(f)(3) only if the replacement candidate pays the filing fee or submits a petition in lieu of a filing fee.

(c) Under the circumstances described by Subsection (b)(2), the appropriate executive committee of each political party making nominations for the general election for state and county officers may make a replacement nomination for the office sought by the withdrawing candidate.

(d) For the purpose of filling a vacancy, a majority of the committee's membership constitutes a quorum. To be nominated, a person must receive a favorable vote of a majority of the members voting.

(e) A vacancy in a nomination for a district, county, or precinct office that was made by primary election may not be filled before the beginning of the term of office of the county executive committee members elected in the year in which the vacancy occurs.