Drinking around the world is pretty fun. I go to Disney World just about every year. We have attempted to drink around the world several times, but it takes a LONG time. The only time we were successful was on my honeymoon. You need to plan on it taking much longer than you think, and it's best to share. I didn't really get into craft beer until after my last time in Disneyworld, so I can't say much for what was available. You have piqued my interest though, as I plan on going again in January.
I've always stayed on property and fly, so I've never had a car. One thing many people in my situation do is use a grocery delivery service, like gardengrocer.com. Looking at their website though, they don't really have much craft beer. Mostly BMCs, but some Dogfish Head, one Red Hook, and one Florida Beer.
If you are just looking for a pint in the park, Beers and Ears
http://beersandears.net/wdw-beer-list/ has a good list of what's on tap. They even have an app to find beer near you (iPhone only).
As GilSwillBasementBrews said, there is a brewery in The Boardwalk resort (just a short walk/boat ride from the International Showcase in Epcot) called Big River Brewing Company. Here is an article about it,
http://www.today.com/food/where-dad-disney-gets-brew-oasis-craft-beer-750875?franchiseSlug=foodmain. I'll definitely be trying this out on my next trip.