Honestly, this runaround should be reason enough for people to file a chargeback, especially when they say something like the promo being a lottery which very much contradicts the terms and conditions of the promo on Grubhub. If I didn't get my emote, I definitely would be considering filing a chargeback if I got the runaround from Grubhub's customer service.
The codes themselves are sent by grubhub, though SE would likely be the person that supplies their codes for obvious reasons. Also, a limited supply doesn't really make any sense considering that it is a digital item.