I’d say there are about a million ways to easily do this. Ideally, you don’t want to use a dedicated app that people need to search for, download, and install because getting them to do this will be a pain (speaking from professional experience as a HCI researcher). Something people already have on their phones will definitely work best.
The simplest solution, of course, would be good ol’ SMS – print a sign saying TEXT YOUR REQUEST TO XXX-XXX-XXX and hang that outside the booth. If you’re not comfortable handing out your number like that, use Twitter – again, print a sign saying TWEET YOUR REQUESTS – USE HASHTAG #REQUESTSTODJ or something along those lines, and then follow that hashtag yourself. Or Facebook, create a page for the event you’re playing at and let people write requests on the wall.
I’m sure there are tons of apps out there that let you do many-to-one messaging, but like I said, getting people to download and install anything is usually difficult, especially due to the huge variety of brands and models out there – you don’t want to restrict this service to those with, say, Android smart phones.