If the person has damaged your car in a hit and run you hopefully first called the police so that law enforcement could write up an accident report. If you know who the offending party is it would appear that the hit and run driver was caught or found after the accident. Normally police would then try to get the at-fault party's insurance information and place it on the report.
The Texas insurance regulator site notes that if another driver refuses to tell you the name of his or her insurance company, get a copy of the police accident report. The accident report should list the other driver’s name and insurance company. If the police did not investigate the accident, you can report the driver’s refusal to police. This could result in a report identifying the driver’s insurance company.
In addition, the Texas Department of Public Safety keeps files of forms that show the insurance companies of people convicted of DWI or driving without insurance. You can try calling the DPS customer service Bureau at 512-424-2600.
If for some reason the police have not been involved yet you should contact them now and they may be able to help you find out about the hit and run driver's insurance situation. As you are likely aware in Texas there is now a Vehicle Insurance Verification system in place that the TX DPS and Department of Insurance refer to as TexasSure.
TexasSure, the TX financial responsibility verification program was created by Senate Bill 1670, which added Subchapter N, Chapter 601 Transportation Code. Section 601.452 of the Transportation Code, as added by SB 1670, provides that the Texas Department of Insurance (TDI), in consultation with the Texas Department of Public Safety (DPS), the Texas Department of Transportation (TxDOT) and the Texas Department of Information Resources (DIR) “shall establish a program for verification of whether owners of motor vehicles have established financial responsibility.”
This new program is designed to reduce the number of uninsured motorists in this State; operate reliably; be cost-effective; sufficiently protect the privacy of the motor vehicle owners; sufficiently safeguard the security and integrity of information provided by insurance companies; identify and employ a method of compliance that improves public convenience; provide information that is accurate and current; and be capable of being audited by an independent auditor.
TexasSure users include the DPS, sheriffs, local police, and vehicle registration offices. These users will be able to obtain accurate and timely insurance information on a given vehicle and/or driver promptly upon request. For example, after the system is implemented law enforcement officers will be able to electronically confirm whether or not a registered vehicle or motorist has insurance coverage in effect.
This verification system is supposed to be rolling out gradually in TX so check with local law enforcement officers to see if they can help you look up and see if the hit and run driver is insured so if he is you can make a claim for your vehicle's damages. If the driver ends up not being insured then you will likely have to take him to court and seek a judgment for the cost of your repairs.