There is no limit in the Constitution that prohibits individual US states from exchanging representatives with foreign countries or from expressing or sending support to them. However, there are some caveats, of course, and it’s a very nuanced area of law that has interesting implications:

1. Accepting formal diplomatic representatives from another power is deemed under international law to mean recognising the independence and sovereignty of the power whose representatives you are accepting. Which essentially precludes formal diplomatic ties from consideration. This is why the US doesn’t accept diplomats from the Republic of China (a.k.a. Taiwan) and refuses official Taiwanese diplomatic and service passports, but is more than happy to accept “unofficial” representatives.
2. Any representatives sent would not have the power to contract treaties as US states are not competent under US law to enter into treaties or make any other binding obligation to other countries. This is problematic because that means they can’t even do as much as rent an office space in another country without the involvement of the US federal government.
3. The primary reasons that a country might consider hosting a diplomatic mission of a foreign power is so that they can (1) complain to the ambassador about that foreign power doing things that they don’t like, (2) so that the foreign power can issue passports and visas within the host country, (3) so that consular services can be provided by the foreign power to its citizens or subjects living within the host country, and (4) negotiate treaties. Since US states don’t really do anything abroad that can’t be handled or complained about through the US Department of State, and because US states don’t issue passports or visas, and because consular services to US citizens is already provided through the diplomatic missions of the United States, it is unnecessary for any country to consider hosting a US state diplomatic mission.