Here is a recipe for mayo that doesn’t contain eggs, nor is it made with tofu (the recipe for tofu mayo is here).
Vegan Mayonnaise Recipe
3/8 cup (6 tbls) fresh-squeezed lemon juice
1 cup soy milk or coconut milk
1/2 tsp sea salt
1/2 tsp paprika
1/2 tsp Dijon mustard
3/4 cup canola or olive oil
Put all the ingredients except the oil in a blender. Blend on the lowest speed. Gradually – just a drop or two at a time – add the oil until the mayo begins to thicken. Keep blending until it’s thickened and smooth. Transfer to a jar and store in the fridge.