If I take the standard deduction on my 2018 federal return, can I itemize on my 2018 Ohio return?

Simple answer: yes.

But, Ohio doesn't really have itemized deductions. They do allow a medical expenses deduction that shadows the federal medical deduction and a handful of other schedule A deductions that do not resemble any federal itemized deduction.

All Ohio filers get the Schedule A deductions (and additions), regardless of whether they itemized or used the standard deduction on the federal return.