Use fresh ingredients. The flavor of the garlic and lemon are the stars of this recipe, so be sure to choose real, fresh versions of both.
Use your favorite vegan mayo if you like. Any type of prepared mayonnaise can be used here.
It's important that the garlic is ground very finely for aioli. The easiest way is to use a fine rasp grater, like a Microplane. You'll need this tool to grate the lemon zest as well.