Access to another individual's DMV records, including their license number, is restricted by the federal Driver Privacy Protection Act. This means that in Texas (and every other state), you must have "permissible use" as defined by Section 2721 (b) of the DPPA in order to access those records. If you have a reason for searching someone's driving records that is valid under the DPPA, you simply have to fill out a form, pay a small fee and mail in your request. Records requests take up to 14 days to be processed and returned to you.
Read the DPPA (see Resources). Section 2721 (b), titled "Permissible Uses," provides the exceptions that can grant you access to another person's DMV records in compliance with the DPPA. Check to see if you qualify under any of those exceptions. Note Section 2722 (b) on "False Representation," which states "It shall be unlawful for any person to make false representation to obtain any personal information from an individual's motor vehicle record."
Go to the Texas Department of Public Safety's website (see Resources). Click "Tx Driver License," then "Record Information," which will lead you to the records request page.
Download the DR-1 Application for Copy of Driver Record (PDF) found on the page. Print and fill out the form, including the second page, where you must indicate the exception to the DPPA that gives you "permissible use."
Mail the form to the address noted, making sure to indicate the type of records check you desire, and include the appropriate fee with the filled-out form. The records request will be processed, and records will be sent to you within two weeks.