The Alliance Defense Fund has filed suit against a New Hampshire school district because they allow non-profit organizations to distribute flyers through the schools but won’t let religious groups do so. The lawsuit is correct, of course; if a school is going to allow community groups to access their facilities or distribution systems, it cannot shut out religious groups from the process. I expect this case will be settled rather quickly.
If you really want to stop such distributions, it’s rather simple: try and distribute a flyer for a local atheist gathering. The parents and teachers will freak out about it and the school will do away with all such distributions. That’s what happened in a school in Virginia when one group submitted a flyer for an atheist summer camp. They’ll do away with such distributions rather than let the Evil Atheists use it too.