Hmmm, well basically, what you need to do is to set up an IPSec tunnel (this is what will get you through you school security measures) to "some" network that has the ability to get to this League of Legends game.
The easiest way to do this is to find a friend that has a home router capable of terminating an IPSec tunnel. Then, connect your PC to the school network, and then once you have a good connection, use some sort oof additional VPN software to connect to your friend's home router.
http://www.howtogeek.com/68061/setup-ssh-on-your-router-for-secure-web-access-from-anywhere/
I know you said you can't do "port forwarding", but in essence, you cannot do what you want without tunneling through the school network, unless their administrator opens up the access for your charity event.
You can try an IPSec VPN (all traffic goess over TCP port 22 to the home router, and then on to the LoL servers, or you can try an SSL VPN (TCP port 443) if the school network blocks port 22.
http://www.acevpn.com/knowledge-base/configure-ace-ssl-vpn-on-dd-wrt-router/
That's it. No other options. AND.......if your school's IT department is sophisticated enough to do deep-packet inspection, they will see the tunneling and block it as well. (but doubtful they are that advanced.)
Sorry, man. whether it's for charity, or to play LoL in Mr. Baumgartner's 4th period Economics class, it's a fairly technically challenging thing you need to do. I've given you some good info to start, but you need to hammer out the details, tho.
On a positive note: If you can figure out how to do stuff like this, you are well on your way to a networking career. :-)