I found this fantastic recipe for mayo using aquafaba on Minimalist Baker. You only need 1/4 c brine from a can of chickpeas but it whips up nicely like the consistency of eggy mayo.

1/4 c aquafaba
1/2 tsp salt
1/8 tsp cream of tartar
1/4 tsp ground mustard
1 tbsp lemon juice
1 1/2 tsp apple cider vinegar
1 tbsp brown rice syrup or maple syrup
3/4 c vegetable oil (neutral flavour such as rice bran, not olive oil)

Put all ingredients in a bowl except for the oil. Whisk with an electric hand beater until combined. Then slowly trickle in the oil and keep whisking at high speed until fully combined, thick and smooth (can take up to 10 minutes). Taste to see if you need a little more salt/sweet/tang and adjust accordingly.
Use immediately or chill in fridge until cold. Keep in a sealed container in the fridge for up to a week.
I used my mayo in a potato salad with capsicum, gherkins and parsley.