Put the aquafaba, vinegar, mustard, garlic and salt into a small bowl or the narrow mixing jar that comes with a stick blender (that's what I used).
Blend briefly to incorporate everything.
Blending the whole time, start slowly (very slowly) drizzling in the oil. Like magic, after a while the mixture will start to thicken and become glossy. Keep adding oil until you're happy with the thickness of the aioli (the more you add, the thicker it'll become). I was happy with just over half a cup of oil, and it took about 7-8 minutes.
Taste and adjust salt, mustard or vinegar to please your tastebuds.
I think this will keep safely for about a week in the fridge, though it's not a huge recipe so probably won't last you that long!
Leave out the garlic, and you have eggless mayonnaise.