You don’t need a license to become a Texas process server, but certification is required.
The Texas Supreme Court oversees the certification of Texas process servers, but the actual process is handled by the Judicial Branch Certification Commission (external link to their website here).
And if you want the legalese, it’s here:
Rule 103. Who May Serve
Process—including citation and other notices, writs, orders, and other papers issued by the court—may be served anywhere by (1) any sheriff or constable or other person authorized by law, (2) any person authorized by law or by written order of the court who is not less than eighteen years of age, or (3) any person certified under order of the Supreme Court. Service by registered or certified mail and citation by publication must, if requested, be made by the clerk of the court in which the case is pending. But no person who is a party to or interested in the outcome of a suit may serve any process in that suite, and, unless otherwise authorized by a written court order, only a sheriff or constable may serve a citation in an action of forcible entry and detainer, a writ that requires the actual taking of possession of a person, property or thing, or process requiring that an enforcement action be physically enforced by the person delivery the process. The order authorizing a person to serve process may be made without written motion and no fee may be imposed for issuance of such order. (Amended June 10, 1980, eff. Jan. 1, 1981; July 15, 1987, eff. Jan. 1, 1988; October 7, 2004, eff. July 1, 2005)
Check your state’s website for the most up-to-date information on how to get certified to be a process server in Texas.
Another good resource is the Texas Process Servers Association.