If you
know a hurricane is definitely headed towards WDW and your vacation, further precautions *might* be needed (I "waterproof" my personal
ECV much more because of it's design and because it's MINE) but overall, the best solution is what
@powellrj recommended.
Put on your poncho while sitting on the
scooter. Take the long part that goes down the front, and drape it up over the controls (where you put your hands to steer the scooter) so that someone standing in front of the scooter sees the long part go down over the front. Take the long part that goes down the back, and don't sit on it - drape it over the seat back, and let it go down the back of the ECV, so that someone standing behind the scooter sees it go down your back, and the back of the scooter. (I swear, one of the these days, I will remember to get a picture of this "in action" when we are there - it's much simpler to understand if you can see it.)
If the ECV is a rental, and comes with one of the little clear "shower caps" to cover the controls on the steering, go ahead and cover it, even with the poncho on.
I have, during hurricane season, used a cheap plastic shower curtain to cover my personal ECV when we were inside attractions and rides, but typically - unless there is a tropical storm or a hurricane - Florida showers that pop up in the afternoons, or drift through during the day can be handled just fine with a poncho. If you need to cover the ECV while you are away from it, you will need a way to prevent it from blowing off the scooter - so bring some binder clips if you are really worried.
(a Parks poncho works great for this - as does one of the little el-cheapo Coleman ponchos you can pick up at Target or Wal-Mart)