Yes, you can claim a credit for OH local income taxes as long as the income is subject to MI tax.

From the MI Department of Treasury website,  Credit for Income Tax Imposed by Government Units outside Michigan:

Line 18a: Enter the total income tax paid to other government units. If you paid tax to more than one unit, attach a schedule showing the tax paid to each government unit.

Include the amount of income tax paid to:

• A nonreciprocal state (Any state other than Illinois, Indiana, Kentucky, Minnesota, Ohio, or Wisconsin)
• A local government unit outside Michigan, including tax paid to local units within reciprocal states (e.g., City of Toledo)
• The District of Columbia
• A Canadian province.

You may not claim a credit for taxes paid on income not included in Adjusted Gross Income (AGI) or subtracted on Michigan Schedule 1 (e.g., rental or business income from another state, part-year resident wages, etc.).