In most cases, you do not need to carry any proof of immunizations for travel to China, Hong Kong or Japan. You also do not need proof of immunizations to apply for a visa.
One special exception is if you have traveled to certain countries within Africa or South America since you last renewed your passport. If you carry a passport from those high risk countries or are arriving from a flight from those countries, it may be safer to carry proof of immunization. While it is not unheard of to be put into quarantine in China for simply having a passport stamp from those countries, it is in practice, rare for the person stamping your passport to be paying enough attention.