I think you're probably going to need to drive. Disney World is sort of an enclave, there aren't any trails or walking paths to get in and out of Disney property. Even getting from a hotel to a park or another resort generally means driving. I don't think there are any great birding locales within walking distance of any of the Disney resorts.
So if you are willing to drive half an hour or so, you've got some great birding locations in the vicinity. Check out Orange and Osceola Counties on the Florida Birding Trail:
This isn't to say you can't see great birds on Disney property, within walking distance, because there are really cool birds all over the place in Central Florida in April. If you take a walk around your resort in the morning there will undoubtedly plenty of warblers, flycatchers; northern mockingbirds, grackles, blue jays, cardinals are ubiquitous. In the retention ponds and canals on Disney Property you can see pretty much any kind of Florida herons and egrets (except Reddish Egrets). Just watch out for gators! And I've had good luck spotting birds at the parking lot of Epcot, where they have several lines of trees between the segments of the lot. And you'll see red shouldered and red tailed hawks atop street signs, kestrels and shrikes on power lines, as well as swallow tail kites in the sky.
So you shouldn't have a problem seeing 20-30 of the most common species of Florida birds right near your hotel. But if you want to see some of the less common species you'll need to drive a bit out from a Disney property.
I'd be remiss if I didn't recommend Merritt Island. It's an hour or so drive from where you will be, but well worth it if you've only got one day for Florida birding.