I read about the state of Missouri & Kansas withholding tax refunds of undocumented workers and inquired of the Ohio Dept. of Taxation. Here's most of the reply -

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Two important developments within the recent years have spurred the undocumented workers to file: (1) the IRS has authorized certain tax preparers to act as "acceptable agents" to help the IRS issue ITINs; (2) the refundable Additional Child Credit of $2,000 can be based on children living in Mexico, the United States or Canada. With these two developments, a worker who would otherwise have a very small refund or balance due on minimal wages is often entitled to a $2,000, or more refund. Certain preparers help undocumented workers obtain ITINs under questionable circumstances, and then file returns which show the undocumented workers' real names and ITINs, but are accompanied by W-2s which show the names and SSNs that the workers used when applying for jobs. Obviously, the tax preparers are highly motivated by getting a significant share of the refundable Additional Child Credit.

The State of Ohio has a similar policy ( as Missouri & Kansas) regarding discrepencies between ITINs and W-2s forms. When an Ohio return is received which has an ITIN accompanied by a W-2 which has a different name and social security number, the Department of Taxation sends a letter to the taxpayer asking for verification of the taxpayers's identity by submitting a corrected W-2 issued by the employer showing the correct name and udentifying number. The taxpayer, especially if they are an undocumented worker, cannot supply this information so the refund is withheld.
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I added the bold type.

So I wonder, if the State of Ohio follows up with the employer, regarding any of these discrepencies ? I doubt it. I think I will inquire as to their responsibility to notify the employers.