You can check in for your SWA flight at Guest Relations at any of the theme parks or Downtown Disney, or at the Lobby Concierge desk at any WDW resort (even if you're not staying there). Just don't forget that you need your confirmation number - write it down in a notebook, in the back of your autograph book, on a slip of paper in your wallet or lanyard, or note it on your cell phone.
You can also check in for your SWA flight using any web-enabled mobile phone. Even if you don't have a data plan, most mobile phones can access the net, and you'll incur per-minute or per-megabyte charges for the few minutes you're online. When checking in by mobile device, you obviously won't be able to print your boarding pass until you get to the airport, but don't worry; printing your boarding pass at the airport will not change your boarding group or number.