An Arkansas contractor taking work in Ohio deals with the same four questions everywhere: who licenses the trade, when a licence is actually required, when workers compensation kicks in, and where the business registers. Below is Ohio's answer to each, with the agency's own link.
Licensing
Ohio Construction Industry Licensing Board, Ohio Department of Commerce
No statewide general-contractor license; GC licensing is municipal. OCILB (Department of Commerce) licenses electrical, HVAC, refrigeration, plumbing and hydronics contractors statewide under ORC Chapter 4740.
Workers compensation
Ohio Bureau of Workers' Compensation
Monopolistic state fund: employers with one or more employees must obtain coverage through Ohio BWC's state insurance fund or be granted self-insurance status.
Business registration
Ohio Secretary of State (Ohio Business Central filing portal)
Tax registration
Ohio Department of Taxation
https://tax.ohio.gov/business/register-for-taxes/business-registration
New hire reporting
We could not confirm a new hire reporting site.
Most states require newly hired employees to be reported within a short window of their start date.