Can we withhold her final check until she turns in this paperwork? Nope. You cannot hold an employee's wages for any reason and you sure can't hold their paycheck hostage in order to try and strong-arm an employee into signing a non-compete agreement on their way out the door. While Ohio has no specific regulation regarding when final pay is due, the courts have generally held that final wages must be paid no later than the employee's next regularly scheduled pay date.
We are in Ohio and have an employee who just resigned. She is refusing to sign and turn in the non-compete agreement, which was implemented during her employment with us. And what inducement have you provided for her to consider signing such a document? If you want to induce her to sign the document and have it be valid, you need to provide some "consideration" over and above that which she would normally receive (such as her paycheck.) In other words, if you want her to sign the non-compete, you need to cough up some $ and in an amount sufficient that she'll want to sign.