The main difference between regular mayonnaise and Japanese mayonnaise is that the latter is typically made with rice wine vinegar, which goes much better with the flavours used in the sushi rice and other sauces found with sushi.
The simplest solution is to find Japanese mayonnaise at your local Asian grocer. The most popular brand is called Q.P. or Kewpie (the non-descript bottle pictured above).
In a pinch you can use store bought real mayonnaise, but don't use Miracle Whip (even if you like it on your sandwiches, it is a salad dressing that doesn't mix well with sushi flavours).
If you are feeling ambitious you can make your own mayo from scratch. Take your favourite mayo recipe and substitute rice wine vinegar instead of white wine vinegar. I've done this on a number of times using a variation of this Gordon Ramsay mayonnaise recipe to which I added 1 tsp of rice wine vinegar.