No, secret cults do not make it very far in the modern world. They are too secretive, bizarre, and creepy for the most part. Modern establishments and individuals will not stand for it. If the cult has devious plans, it will get caught and stopped. If it has good plans, it needs to restructure them in such a way that they'll be accepted.
The best way to think about this issue is to compare it to similar situations. ISIS (The Islamic State of Iraq) is essentially a religious cult. They've caused some havoc all around the globe, but nothing they have done has accomplished their goal. The same can be said for organizations in the United States, like the Ku Klux Klan. They want racial and sexual inequality, but they've had no success. People have come too far. People have gotten too used to idea that we all can coexist. Because there are too many people in this world who are willing to fight against injustice and inequality of all kinds, the Japanese cult cannot succeed.
The history of the world has proven that one group and/or person cannot and has not succeeded in complete world domination and imposing their will on all others. Other societies and groups will always stand up for their own beliefs and do what is right to deny this from happening. So, this cult in Japan will not succeed with their plans.
Nippon Kaigi is a conservative Shinto cult in Japan, whose aim it is to return Japan to its pre World War II "glory". There are supposedly many powerful people involved in this organization. With the upcoming vote in Japan, they could win enough seats to enable them to change the constitution. I really do not think that this is possible or even probable, considering some of the things they would change would be to end sexual equality and limit human rights. I can't see modern people of any country voting against these things.