No. Texas bill HB 1760, if enacted in that form would be void on its face.
This law would be void on its face because the federal government has the exclusive right to discriminate based upon immigration status and in the absence of a federal law expressly authorizing state governments to enact such legislation. This is because it is a regulation based upon immigration status where federal law occupies the entire field and pre-empts any and all state legislation.
Similar laws, for example, related to landlord-tenant arrangements by illegal immigrants in Texas, have been held to be void. See, e.g., Villas at Parkside Partners v. City of Farmers Branch, No. 3:06-CV-2371L, 2007 U.S. Dist. LEXIS 44234 (N.D. Tex. 2007) (granting a preliminary injunction enjoining Farmers Branch Ordinance 2903 adopted on Jan. 22, 2007).
A detailed analysis of federal pre-emption of state and local immigration laws with lots of supporting legal citations can be found here.
Why Propose Such A Law?
The legislator proposing the law almost surely knows that his bill is unconstitutional on its face, but it simply introducing it to make political points and as a model for Congress, which would have the power to enact such a law.
It is also far less likely, although possible, that the legislator has a good faith belief that the federal courts might overturn existing precedents to uphold this law.
Interstate and International Commerce Pre-Emption
There is also some pre-emption of state authority to regulate interstate and international commerce as a result of the "dormant commerce clause". But, that pre-emption is not nearly so complete, because the foundation of most commercial transactions involves the application of state law governing contracts and business dealings between private parties. (The dormant commerce clause actually applies with more force to discriminatory interstate regulation of commerce than international regulation of commerce and is also bolstered by the two "privileges and immunities clauses" in the United States Constitution as amended.)
What Could Texas Do?
Texas probably could enact a law that would tax people who wire money anywhere to pay a 3% fee plus $5 per transaction, because state taxation power is basically plenary and unlimited, so long as it isn't structured in a form that is pre-empted by federal law (as in the case of HB 1760) or unconstitutional (e.g. poll taxes are prohibited).
If it were only applied to journalists, it would violate the First Amendment.
If it were only applied to out of state transactions, it would probably violate the dormant commerce clause unless it met a heightened rational basis level of scrutiny or some sort of reasonableness standard.
The dormant commerce clause would also probably violate it if it applied to all international transactions, but this would be a closer call.
There might also be pre-emption from existing federal legislation regulating or taxing the same transactions, or from bilateral or multilateral treaty obligations that limit the imposition of such fees (at least for wires to some treaty party countries) such as the North American Free Trade Agreement (NAFTA) and the General Agreement on Tariffs and Trade (GATT) which is managed by the World Trade Organization (WTO).