Foreigners can have ownership interests in U.S. businesses. In most states there is nothing to prevent direct ownership. I have set up several LLCs for real estate investors. There are usually no problems with formation of the entity. The complications come from complying with the tax laws, as there may be requirements to withhold income taxes from any distributions to foreign owners. You will have to get an individual taxpayer identification number (ITIN) for foreign owner(s). The rates of withholding will be dependent on any treaty agreements.
If you can form an entity in Delaware, and I'm not aware of any issues that would prevent it, then your Delaware entity can register with the Texas Secretary of State as a foreign business entity and conduct business in Texas.