You can file multiple Ohio School District returns. There was a limit of 1, but that was several years ago and it was a ProSeries software limitation, not Ohio's system.
With multiple School District returns, due to a client moving or the employer's incorrect withholding, I always paper file all the returns with the W2s and anything else I think they will need. I prepare a lot of these multiple SD100s
In my experience, e-filing multiple SD100s usually generates a notice from the Ohio Dept. of Taxation requesting copies of W-2's to verify withholding, along with leases, County Auditor's tax bill or utility bills as proof of new address, if client moved.
The notice goes to the taxpayer, of course, who generally does nothing or brings in the notice next year, if they remember. By that time Ohio has issued a Billing Variance Notice, also ignored by the client, and Ohio starts offsetting the State refund to pay for a School District tax from a prior year because a response and documents were not received for the original notice. Then it's transferred to the Attorney General's office for collection and becomes one BIG mess. It's much easier to do the work and paper file the returns with the documents I know Ohio will request.
I also file Non-Resident returns for clients with a new filing address but no School Tax liability till the following year because Ohio sends a notice, based on the new address, that a return has not been filed for that School District for the current tax year. Ohio School District's are a royal PITA and I'm surrounded by them. 🙂