Yes. There is no requirement that you have car insurance to file an injury claim. If the other person is at fault, then their insurance company is required to pay for your injuries and expenses.
When it comes to auto insurance, Texas uses what is called a "fault" system. This is a system that requires drivers to pay for the accidents they cause. There is no requirement that you have car insurance. Simply put, if another person is responsible for causing the accident, they (or their insurance) should pay.
A word of caution. Going without insurance, even if not technically required for you to file an injury claim, is a bad idea. The other party could claim that you are at least partially responsible for the accident. If a jury agrees, you might have to pay for some of the injuries or expenses. Also, having uninsured motorist insurance would help protect you if the other party’s insurance coverage is insufficient to pay for all your injuries and expenses. (See: What Is the Difference Between Uninsured and underinsured Motorist Coverage?) Finally, driving without insurance is against the law in Texas and if caught, you might get a ticket for driving without it.