What's the meaning of state here?
"The law requires health care providers to request a patient state in writing that they received the notice at their first visit. Often, that means the practice will ask a patient to sign a form stating that they received the notice that day."
Is it a verb or a noun?
It is a verb meaning "tell" in a formal way.
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Think of it this way.
"The law requires health care providers to request [ that ] a patient [ should ] state in writing that they received the notice at their first visit. . . . "
Now that I think of it, "requires" eliminates the necessity of "to."
Because of the subjunctive, shouldn't the sentence be:
The law requires that health care providers request...