An employer's ability to even ASK employees to take a polygraph test is extremely limited. A federal law, the Employee Polygraph Protection Act, severely limits an employer's ability to use polygraph tests. Some States have similiar laws which make it all but impossible for an employer to request polygraph testing, much less have a polygraph test performed.
I think your husband should do some on-line research on the EPPA and contact his State's Department of Labor to inquire whether Ohio has a similar law before this goes any further and your husband is advised on his legal rights to refuse to take the test.
Personally, I would not submit to such a test (even though I was 100% innocent) unless I was absolutely convinced of the expertise of the person administering the test and perhaps not even then. I think your husband needs to research his legal rights before deciding whether he wishes to agree to this. A poorly administered test could result in his losing his job and being accused of stealing. He has a lot on the line here.