(3-30-22) A new referendum item was given tentative approval by the OHSAA Board of Directors during a Special Meeting on March 19th. OHSAA member Principals will now have another item to consider and vote on.
The new option will pertain to public schools. Private schools already have that opportunity thanks to the Ohio Revised Code (see below).

EX: Coldwater does not offer boys soccer, Celina does. Student-athlete, if referendum passes membership vote, would have the opportunity to play for Celina
Below is an email reply to SSN on the new issue and the current one that already allows Private school students the opportunity to play at a public school-
From Kristin Ronai, OHSAA’s Director of Compliance–

Thanks for checking. Yes, state law (ORC 3313. 5311), and thus Bylaw 4-3-1, Exception 6, allows non-public school students the ability to play sports for their residential school with no enrollment requirement. There is no comparable state law for students attending public schools which is why this referendum is being proposed for a membership vote. Obviously the benefit of it passing would be that it provides students attending a public school with additional participation opportunities. However, there are also concerns about implementing this type of a proposal which have been expressed by our member school administrators.
More details about the pros and cons of such a proposal are outlined under Issue 4B on pages 4-6 at this link: https://ohsaaweb.blob.core.windows.net/files/SchoolResources/refvote/FINAL-2022ReferendumItems.pdf
And here is a link to a survey of member school principals we conducted earlier this year:

We will be explaining all 14 referendum items at six in-person meetings throughout the month of April. Then the principals will have the final say on the matter during the voting period between May 1-16.
Presently private schools already have the option if their school does not offer a sport to compete at a public school (see below).
That happened this school yearLima Temple Christian did not field a girls basketball team, that allowed one player to play for her public school Lima Shawnee. The student-athlete also competed in volleyball and in the spring (currently) back at Lima Temple Christian.
Just as an example, Shawnee does not offer girl gymnastics, but Lima Sr. did…under the current ruling the Shawnee student-athlete cannot compete with Lima Sr. team.
The Most Commonly Asked Questions for Non-Enrolled Students
There are students who attend non-public schools in Ohio eligible to participate if their school does not offer a specific sport?
A student who attends a non-public school in Ohio, whose parents reside in Ohio, shall have a participation opportunity – not a guarantee – at the public school the student is entitled to attend pursuant to ORC 3313.64 or 3313.65. A student who attends a non-public school may also have a participation opportunity at the public school located in the school district in which the non-public school is located (if different than the residential district) subject to the mutual agreement between the superintendent of the school district in which the student is entitled to attend and the superintendent of the school district in which the student is seeking to participate. Of course, this participation opportunity applies ONLY if his/her non-public school does not sponsor the specific sport. The student must be eligible in all other aspects (scholarship, age, transfer, semesters, etc.). If the public school in either the parents’ district of residence or in the district where the non-public school is located does not offer the sport, then the student simply has no participation opportunity.
Once a student has commenced participation either at the public school located in the district of residence of the parents, or at the public school located in the district where the non-public school is located (should those two schools be different), then that student’s eligibility to participate in those sports in which the student’s non-public school does not offer shall remain at that respective school for the duration of the school year, regardless of what sports that school may or may not offer. For the subsequent year, should the student decide to transfer his/her eligibility to the other school at which they also have a participation opportunity, no transfer consequence shall apply in accordance with the ORC.