What kind of beach are you looking for? Do you want rustic, something with activities, or something in-between? Will you need to rent chairs/umbrellas, or will you have your own vehicles to bring them from home?
Be aware that the currents on the east coast (especially the Treasure and Space Coast areas) can be rather more treacherous than on the west coast, you will be dealing with colder water and rougher surf than in the Tampa area. When are you going? (There have been jellyfish issues on the east coast lately at certain times of year, just as there have been red tide issues on the west coast.)
The very closest beach is Cocoa, but it might not be the best choice for you, depending on what you are looking for.
You *could* do Cocoa, Canaveral or even Daytona Shores as a day trip, but it would be a long day that involves getting up very early. You have to go through Orlando's commuter traffic to go to those beaches from WDW, so you'll have to try to beat the morning rush. I always recommend a weekday for a daytrip, because the parking will be easier -- the locals come out on weekends, putting spaces at a premium. It will be about two hours until toes-in-the-water for most people, so I would think you would want at least 4 hours on the beach; making it roughly an 8-hour minimum excursion.
