I worked in Payroll for a large, nationwide company. They may say they are being green, but they save literally millions of dollars by not giving you a check stub. Well, it depends on how big your company is. I was thinking there was one state that we had to print stubs for, but I can't remember. I am thinking Rhode Island? Company also forced everyone on direct deposit, another huge money saver for them.
Ok, according to the Dept of Labor, no state is required to provide you with a paper stub. But they must provide access to your pay information. So no, Ohio does not have to give you a check stub, but they should have the proper means of letting you print your own off on the job site.