I guess if it was that easy the schools wouldn't be having the budget problems to start with.
Oh, I didn't mean to be discouraging!

Most businesses are happy to support the school, and an advertising deduction is as good or better than a charitable deduction for businesses. (For individuals, you need it to be a charitable deduction.)
In my area, the big issue with finding outside funding sources is avoiding stepping on anyone's toes. If Big Corporation X normally supports Activity Y, and you solicit them for Activity Z, Y is going to be ticked off. Or if Corporation X normally provides school supplies for Low Income School W, I wouldn't want them to stop doing that in order to sponsor chess club for my kid's Middle Class School V, you know?
I have found that if someone is enthusiastic about an activity, and is willing to contribute significant time, funding tends to get found, one way or another.