Legal Question in Immigration Law in Ohio

No, you can't.

You have to be eligible for adjustment of status in order to submit application for a work permit.

When your wife is a USA citizen, then you can apply for adjustment of status and a work permit (assuming that she had never petitioned for you and the priority date is not current yet).

At this time, you are a F1 student, as I understand.

If you'd like to schedule a consultation, please email at Attorney [@] law-visa-usa.com

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Based on the facts you've presented, your wife cannot and should not apply for you and your permanent resident status. You may be eligible for work authorization pursuant to your F-1 status if you can show extreme financial hardship. Additionally, upon completion of your degree, you will be eligible for OPT and possibly a STEM extension. Hire an attorney to strategize with you and your wife accordingly.