Yes, an electrical outlet is required for islands. If you are on slab, then only good way is to trench the slab for the wire.

However, an outlet isn't required for a mobile cart. 24"x36" is small for an island, but perfect for a cart. Don't forget that you need a minimum of 36" all the way around it for proper clearances for an island as well. For a cart, because it's mobile, it can shift back and forth to give you more clearance as needed.

I just purchased a Boos kitchen cart via Craig's list and it is great to be able to move it around. They have all sizes of carts.
It does have an outlet strip under the 5 inch butcher block top which gives even more options.
If you stay with the island - you can also go with Mocket sockets - they pop up to use. (Not sure I spelled it correctly)