You haven't received an answer to your question because there isn't one. This is a non-starter. Employers will not give access to their employment records to political activists for fear of privacy lawsuits. Ain't gonna happen. Even if you get some to work with you, how do you know the documents the employer used in the hiring are not forgeries? Expend your energies on some of the other suggestions made on this thread.
One idea that just came to mind is; When shift change time comes around, stand outside the plant with a sign asking, "Do you work with illegal aliens? Would you rather work with law abiding co-workers? Stop and we'll talk about making changes."
I wouldn't be surprised if you get swamped in some industries.