Add the soy milk to the bowl of a food processor. Process for about one minute or until the liquid begins to thicken.
Add the vinegar, salt, mustard, and lemon juice to the bowl, and then process for another 30 seconds so that everything is well blended.
Scrape the sides and bottom of the bowl. Then, with the food processor running, slowly add the oil in tiny drops until about a quarter has been added (this is critical for proper emulsification).
You can be a little less strict when you notice that the mixture is beginning to thicken and emulsify. Continue to add the oil slowly, but increase to a thin stream instead of drops of oil.
After adding all of the oil, scrape the bowl, and then process for 10 seconds.
Taste the mayo, and then adjust with more salt or a sweetener if you feel it needs it.
As the mayonnaise sits in the fridge, it thickens even more. If you think the mayonnaise is too thin, you can blend a little more oil into it. Do this slowly.