I think your analogy is a bit flawed. The local Boy Scout troop would love to have such a promotion if the rewards were achievable. 10,000 sandwiches to earn one free sandwich is obviously ludicrous but if it was more like "When the group buys 100 sandwiches, Subway will cater your next pack meeting", then the kids/parents would be all over that.
Pizza Hutt used to do something similar with school programs back in the South East US. Kids that participated in a reading program got a free personal pizza. I forget all of the details since i was in 2nd or 3rd grade at the time, but along with bringing more people in for the free part, it also got a lot of people in just because they were supporting a program like that.
You're probably thinking of Book It (http://www.bookitprogram.com/faq/). I don't think it is South East USA specific (we had it in the midwest and the website doesn't indicate that it doesn't apply nationally.)
This totally works — especially if the recipient is an organization or a cause. Grocery stores and other places do this where on certain nights they will donate a percentage of proceeds to a local school, etc.
People are willing to trade a small reward for themselves for a larger reward that a group can achieve.
There's a local deli here in Seattle where you drop your business card in for a chance to get catered or something. (It's been a while since I've been there; too packed.) If all your coworkers drop in, likelihood kinda shoots up.
