Despite the wishes of some in the United States, children born in the United States become citizens at birth, even though their parents are not citizens, and even though their parents are not legal residents of the United States. That’s the law.
Why is there a problem getting birth certificates for these US citizens in Texas? It comes from the fact that not just anyone can apply for a birth certificate. It has to be the person themself or a parent. The parent has to provide ID, and there’s the rub. The Texas Department of Health has begun enforcing a 2008 rule that says a Mexico consular-issued identification is not acceptable. Aaron Nelson of the San Antonio Express-News writes:
A citizen may request his or her own birth certificate with a driver’s license, or a Social Security card and recent utility bill, for example, but to acquire those documents often calls for a birth certificate.