No, your husband cannot get any work permits in Texas or elsewhere as it stands now. You can sponsor him for green card if you are a US citizen, but he will need a waiver. Those are not easily granted. Schedule a consultation with an experienced immigration lawyer to learn the process and see whether or not it's worthwhile trying now.
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A bit more information is needed but he might be eligible for a work permit under DACA. I highly recommend scheduling a private consultation with an experienced immigration attorney. Many here on avvo, including me, offer free consultations by phone or Skype. Good luck!
Texas does not offer work permits. Work authorization in the USA can only be issued by the Federal Government.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
For the work permit, look into the Deferred Action program for Childhood Arrivals, as my colleague suggested. You mentioned that he didn't finish school here in the U.S. However, he can still satisfy the educational requirement for DACA by enrolling now in a GED program.
Longer term, you want to look into the possibility of him obtaining residency based on his marriage to you. Whether he's eligible to do that here in the U.S., or whether he will need to apply for a waiver and process his visa from abroad, or whether he is barred completely from obtaining a marriage-based residency will depend on a host of factors, including the nature and timing of his entries and attempted entries and his criminal history, as well as other issues.