They do so by special permission which may involve a permit. They also are limited on the number they may catch. They were given a grandfather clause because whale hunting was a major part of their culture and cuisine.
In the meantime the Japanese signed a treaty to limit their whaling and are supposed to be able to only kill whales for research. They misuse the research clause and are killing whales for reasons other than research. This allows them to circumvent the number they agreed upon. From what I've read whale meat is falling out of favor in Japan.
I don't particularlly agree with the protestor's methods because it endangers not only the protestors but also the Japanese crews.
One thing can be said though. While I think the protestors are idiots, they at least are willing to die for their cause. It may come a day when we want people on our side who are as willing to give up all in order to save our Republic.
I hope all of these idiot protestors (as you say)die for thier cause. Well, I will settle for rotting in a Japanese jail cell.