If the school is a public school, it is automatically a 501(c)(3), which means donations are tax deductible. I agree with the earlier poster: running the money thru the PTO is just easier than running it thru the school's accounting system. If OM is consistent with your PTO's mission, and it's clearly shown on the budget report and Form 990, and there's no provision in your bylaws that would prohibit it, then your group is fine handling the OM funds. The PTO is the sponsor.
We stopped handling the yearbook money because the PTO had nothing to do with laying out the yearbook and no say in the spending of yearbook profit - it all went directly back to the school. So, we politely suggested that the school handled their own project, including the money collection (and NSF followup if necessary). If the PTO takes ownership of the yearbook project in the future, we will handle the money too, of course.
My husband & I have been Co-Treasurers of our PTO for 4 years, and I am now moving into the President role. There is no rule or law against running the money through the PTO as long as it is well documented. Our school often pays for "small items" out of their budget throughout the year, then bills the PTO at the end of the year. Again, it is well documented, and we have our account audited annually by a CPA, who happens to volunteer his time to the school for this purpose.
Odyssey of the Mind does run through our PTO account. PTO gives OM a small budget each year ($500ish). Any funds they raise are deposited into our acct and we pay their expenses (they fill out fund request forms like any other committee/group). They can only request expenses up to what has been allocated to them.
The goal of PTO - in improve education and support teachers (the short version). As a educational student group, supporting OM falls under our umbrella. I see no problems with this.
I will also state that a very organized OM group in a nearby district got their own nonprofit status. They have their own accts and handle all their own fundraising/expenses. Takes very on the ball and consistent leaders for this to happen.
Check your charter as a non-profit to see if the activity falls within your charter. If it does and it helps the school, just sponsor the event. Don't see any problem.
Our PTO has done some similar things; in fact, we did the LEGO FIRST stuff this year, which I think is similar to OM. However, it *is* listed as a revolving account on our approved budget, and I will be recording the numbers on the PTO's 990.