You can get a Disney Visa Rewards card at any Guest Services inside the parks or at Downtown Disney. You just show them your Visa and they get you a blank card.
I don't know about transferring rewards from one card to another, but you have to call the 800# to activate the new card anyway, so they should be able to help you.
I always wait until I am at Disney to get my card after forgetting it at home once- it's one less thing for me to remember before I leave!
Have a great trip!