[mjn]

A bit of a complex question, depending on how you word it.

The federal government cannot require a state's agents to affirmatively assist it in carrying out actions or duties that the state doesn't authorize. Broadly that's known as the "anti-commandeering doctrine".

But a state also cannot specifically impede the federal government, for example by harassing its agents or preventing their ability to operate. This is true even if state agents claim to be enforcing state law in doing so, because under the Supremacy Clause of the U.S. Constitution, federal law (and its enforcement) trumps contrary state law.

Neither doctrine has its full boundaries or interactions precisely clarified, as you might guess.

[DennisP]

But the question regarded "an entity involved in illegal activities as defined by the US Constitution," not as defined by state law.

Courts have differed, but some at least have said the NSA's actions are unconstitutional and illegal. If, say, a state is in a federal court district that decided that way, then perhaps the state could make a stronger claim that denying services is legal.