Automobile insurance is not required to operate a forklift in Ohio. Only vehicles that can be operated legally on the road are required to have coverage. With regard to accidents occurring due to the operation of the forklift, workers compensation will cover your injuries and any injuries to fellow employees. If your injure someone who is not a fellow employee of the non profit (visitor, vendor ect) that injured person would present a claim to the non profit and their carrier.
You are covered under the non-profit's insurance, if they have any, if you are using their forklift with their knowledge and consent. Automobile insurance won't cover a forklift unless it is a street legal vehicle.
This response does not constitute legal advice and does not create an attorney-client relationship. Legal problems and solutions depend on their unique facts. Due to the complexity and ever-changing nature of the law and to the limitations of this forum, this information may not be complete or adequate for your specific situation. Laws and regulations often differ from one jurisdiction to another. Mark E. Barbour practices in the State of Ohio.