Yes and no.
For ordinary lawsuits, Rule 6 of the Texas Rules of Civil Procedure provides that you cannot be served on a Sunday.
However, if the lawsuit requests extraordinary relief involving any of the following: injunction, attachment, garnishment, sequestration or distress proceeding, then citation can be served on Sunday.
Also, no matter the type of lawsuit (ordinary or extraordinary), if service of citation is by publication, rather than by personal or other type of service, then a party may be served on a Sunday. (Service by publication means that a party is served by way of a notice published in a periodical, such as a daily newspaper; so, if the periodical has a Sunday issue, service of citation in that periodical’s Sunday edition is permissible.)
— Bonnie Sudderth, Judge of the 352nd District Court of Tarrant County, Texas