Do undocumented immigrants have a right to state-provided counsel?

Not for immigration issues. The right to an appointed lawyer comes from the 6th amendment to the Constitution which reads:

In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the state and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the assistance of counsel for his defense.”

So it only applies to criminal cases.

Immigration isn’t a criminal situation. It’s a civil process so the Constitutional right to legal representation doesn’t apply.