I made this in my Vitamix, so the instructions refer to the settings for a high powered blender only. It’s possible to make vegan mayonnaise with hand held blenders and other liquidisers, and I wager you will have success in trying to do so.Keep in mind the temperature of the blender’s contents as you go. Let it overheat and you will end up with an oily mass that will not solidify. That said, it's best to start with room temperature ingredients (a little warmth in the oil can even be beneficial). Just know that extreme temperatures will affect the consistency.
Liquidise all of the ingredients but the oil, until the garlic is broken down completely and the liquid is thoroughly blended (10-15 seconds on a medium-low to medium setting should do the trick).
While the machine is running on a medium setting, slowly drizzle the oil into the blender. The contents will begin to thicken after half to ¾ of the oil has been added, and will be the right consistency by the time the last drop falls in.
Refrigerate and use as desired within a week or two.
This makes about 300 millilitres (scant 1 ½ cups).