Height Adjustable Kitchen Island on Casters
While you will likely be exempt from the requirement for counter receptacles for small appliances, provided you have those in other (fixed) areas of the kitchen, you may want to consider providing them for convenience. There are a few ways to do that. The simplest would be to use a 20A floor receptacle with a plug/cord from the island that goes to a junction box in the island where you would have receptacles. You may be able to hard-wire the adjustable top to that or, alternatively, use a quad receptacle (instead of a double) with the adjustable top plugged into 1 of the 4 receptacles, leaving three for small appliances (food processor, blender, toaster, etc.) or other convenience items (cell phone charger).
One of the pros can speak to the specifics - you do NOT simply take an extension cord and hack it to connect to the back of the receptacles. But that is the general way to think of how things will connect.
Consider a Pendant
As noted by Harper, a pendant may be a better solution than a floor receptacle. See How can I provide an electrical outlet at my portable kitchen island? for more details.