Recently, there’s been various kinds of mayo-alternatives out on stands. Depending on which shop you’re at, you’ll see different brands. After trying a few–some of which didn’t have that mayo-richness or flavor, I’ve sticked to Soymayo (www.soymayo.com).
The taste, texture, creaminess is just perfect and it works for anything from sandwiches, dips and simply on veggie sticks. It’s egg-free, water-free and based on soymilk with canola oil and natural sweeteners.
At the veggie fest, you can get them for 700 yen (I know, expensive–but that 700-900 yen is about the range for mock-mayonnaise here)  and also at shops like Meidi-ya (www.meidi-ya.co.jp) and the bakery, Pompadour (www.pompadour.co.jp). At Pompadour, I’ve also seen them selling breads using this mayo.
Then again, you can try making your own too:
1. Put 1/2 cup of soymilk, 1 teaspoon of salt, pinch of black pepper and 80 ml of canola oil in a  blender for a few seconds.
2. Add a tablespoon of vinegar and blend again for a few seconds.