Vegan mayonnaise is surprisingly easy to make, and surprisingly good. In fact, homemade vegan mayo is, in my opinion, better-tasting than many store bought mayonnaises, and has the benefit that if you disagree, you have the ability to change the recipe.
You will need a mixer. I recommend an immersion blender with a whisk attachment. This has the advantage that you can make the mayo right in the jar you are going to store it in. I have never beat mayo by hand, but I hear it's not too hard, if you have a good whisk. You may also want a mason jar, but packaging is outside the scope of this recipe.
• 3 tablespoon aquafaba -- AKA bean juice. A can of chick peas (AKA Garbanzo beans) will usually provide you with enough liquid for at least three jars of mayo, plus some beans.
• 1 tablespoon of vinegar.
• 1/2 teaspoon of salt -- except I find this a bit salty, and use 1/4 t. of MSG. So, salt to taste, really.
• 1 teaspoon of mustard. Yes this sounds weird. No you can't leave it out. It helps with the emulsion of the oil and vinegar.
• 1 cup of oil. Your choice of oil, but vegetable oil or canola oil will work just fine.
Mix the aquafaba, vinegar, mustard, and salt. Then drizzle in the oil, while beating steadily. Keep going until it looks like mayo.
I usually use a bit less than one cup of oil, because if you max out the oil it may start separating before you finish the jar. Other than that, I have found these measurements to be fairly forgiving, and you don't have to worry about being exact or adding in too much aquafaba or what-have-you.
You probably want to make your first jar pretty standard, so you can see what the recipe is like. After that, you may want to experiment. It is easy to add a bit of garlic powder or turmeric after making the mayo, but during the mixing process you might experiment with various oils, different types of vinegar, and maybe even different types of mustard (I have not tried any fancy mustards; you're on your own there). I was a bit disappointed with my attempt at adding some olive oil into the mix, and quite enjoyed using up some sub-standard balsamic vinegar in my mayo, but YMMV (and also, some people look askance at brown mayo).