Almost the entirety of foreign policy apparatus is laid out and reserved in Articles 1 and 2 of the Constitution, and therefore inherent in the Congress and the Presidency. For example the President has the power, with advice and consent of the Senate, to appoint Ambassadors, is Commander in Chief of the Armed Forces. Similarly the Congress can declare war. The treaty power is shared among them.
Article 4, section 4 of the Constitution does allow for this:
The United States shall guarantee to every state in this union a republican form of government, and shall protect each of them against invasion; and on application of the legislature, or of the executive (when the legislature cannot be convened) against domestic violence.
You can see it is limited in scope.
We also have the Tenth Amendment that reads:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
The Tenth Amendment allows states to engage in trade missions with countries. Here is a Massachusetts example. Note that limits on what they can and can't negotiate.
These sojourns certainly are not unusual: Cassey studied a decade-long period starting in the mid-1990s and found about 500 trade missions led by governors nationwide, with more than 40 states launching at least one.
“Governors tend to go to the places where there are already strong relationships,” Cassey said. Business leaders say it’s particularly important that the governor personally participate in the trips, rather than send emissaries.
A governor’s trade mission can close deals and open doors. These trips create deadlines to finalize a business deal or an expansion plan, to ensure the politicians have success stories to unveil during their tour. And a governor, as the state’s most prominent elected official, can line up in-person meetings with hard-to-get political officials and business executives.
Finally you have the Logan Act, which was established to prevent anyone from separately creating foreign policy other than the federal government. Wikipedia has a good enough summary:
The Logan Act (1 Stat. 613, 18 U.S.C. § 953, enacted January 30, 1799) is a United States federal law that forbids unauthorized citizens from negotiating with foreign governments having a dispute with the U.S. It was intended to prevent the undermining of the government's position. The Act was passed following George Logan's unauthorized negotiations with France in 1798, and was signed into law by President John Adams on January 30, 1799. The Act was last amended in 1994, and violation of the Logan Act is a felony.