While any trip to Antarctica requires advance planning, you do not need to obtain special permission, the cruise ship company will do this for the entire boat and its passengers.
Remember, Antarctica is not a country by itself since nobody lives there apart from researchers. Parts of Antarctica are administered by different countries. So the ships get the permissions from the respective administrating country. Having said that, access to the continent is highly restricted. The cruise companies can not land on any beach. Which in part would be pointless. Penguins are not found on any of the beaches and sometimes it is more fun to see those Penguin colonies from a distance. It is an overwhelming landscape that you can enjoy from the boat.