The requirement to provide a check stub of some kind to employees is not contained in Tax Law. For taxes you are required to produce a W-2 that allows the employee to file his taxes, and is submitted to the IRS along with Form W-3 transmittel to report withholding of taxes, SS and MC.
The Fair Labor Standards Act (FLSA) only requires employers to keep sufficient payroll records to substantiate the payment and withholding from paychecks, with no federal requirement to actuall provide a copy to the employees. However,under the FLSA, if an employee asks, you have to give them access to the information regarding thier pay.
The requirement to actually provide the employee with an itemized pay receipt or stub is vested in the state laws.
OHIO does not require pay stubs per se. They require, similar to the FLSA, that records be kept, and that if an employee asks for the information, it has to be provided to them.
NOTE: just on a practicle matter: If I were an employee, I would be requesting this every payday, and that would create a burden od the employer, so that it would be easier to provide the information. If an employer is not manipulating the pay records in a negative and potentially illegal way, full disclosure and providing pay stubs and pay receipts is a critical process that helps to build rapport and quality employee and employer relations.