Most amazing thing a game group did for their community?
It's not a gaming group per se, but rather the local gaming store. Each year Hometown Games does a huge event for the Ballard Center, a community organization in Lawrence, Kansas. There are special tournaments where the entry fee is a donation, special auctions, etc. All told their raise quite a bit of money and other donations throughout that event. One of many reasons why I patronize that store.
In my own gaming group, several of our adolescent members have discussed starting a gaming club at their school to introduce gaming to more people and perhaps find a welcoming venue for gamers who don't have one. Last year one of the younger members hosted a one-off event as part of a wider school function and got eight students to show up and play. The adult members have talked about how we can support them by providing miniatures, terrain, story ideas, etc. I hope that all comes to fruition.