It is possible but you may have to do some looking, perhaps in the older communities (10yrs old or less, it's all relative) or in some that aren't as close to Disney as others. Try Lindfields, Indian Ridge / Oaks, Grand Palms, etc. Look through the Kissimmee listings on vrbo http://www.vrbo.com/vacation-rentals/usa/florida/central-disney-orlando/kissimmee#2543
It should be no problem. For example, here's a link for your dates to available properties at Windsor Hills. It's using the homeaway dot com website, where you can plug in your dates to help weed through what's available.
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