There is wired internet access in every WDW resort room. Cost is $9.95 per 24-hour period, and you are not required to enable it for your entire stay, just whenever you want, in 24-hour increments.
Wireless internet access is only available in the public areas (lobbies, etc) at a few WDW resorts such as Yacht & Beach Club and Coronado Springs (mostly those that have a lot of conventions). Again, it ain't free - it's $4.99 for the first 60 minutes, or $9.95 for 24 contiguous hours.
NOTE: If you need to check in for your flight home while at WDW, you can do it at the lobby concierge desk of any WDW resort. All you need is your confirmation number, and they CMs will do the check in for you and print your boarding pass.
If you fly Southwest, you can also check in via web-enabled cell phone. Be sure to save the URL in your phone ahead of time, and carry your confirmation number with you, and you can check in while riding TTA or watching Wishes. I wouldn't try it on Space Mountain, though.
