There is ample evidence that Japan has the ability to make delivery systems. As for rockets:
Japan Aerospace Exploration Agency - Wikipedia, the free encyclopedia
Avionics and engines:
Mitsubishi F-1 - Wikipedia, the free encyclopedia
They can even make modern jet engines, which is more than China can claim.
Japanese companies and scientists were instrumental in the development of modern fuzzy logic and state-space control systems, in addition, their world-renowned electronics design and manufacturing capability indicates a likely ability to produce terminal guidance systems. In short, your claim is slightly ridiculous. Evidence suggest that Japan has been more than capable of producing delivery systems for nuclear weapons since at least the 70's. If you can accurately put an information gathering satellite into orbit, you can make a ballistic missile.