Line a 9x13 pan or baking sheet with parchment paper. Dust with a thin layer of powdered sugar and set aside to prepare marshmallows.
Add the liquid from a can of chickpeas in the bowl of your stand mixer (see notes). Add cream of tartar and mix on high with the whisk attachment for 6 minutes. It should be very fluffy at this point.
Add the vanilla and beat for an additional 3 minutes.
In a small saucepan, combine the water and agar powder. Bring to a slow boil. Boil for 3 minutes and then add the sugar. Slowly boil for an additional 3 minutes while stirring constantly.
Pour the agar+sugar mixture slowly into the aquafaba mixture while beating with your mixer on low.
Beat for 30 seconds on low until it is all incorporated and becomes slightly shiny.
Immediately pour onto prepared baking sheet or pan, spreading evenly with a spatula.
Sprinkle with powdered sugar if desired (see notes).
Refrigerate for at least 2 hours before cutting into small squares or using cookie cutters to cut into different shapes.
Notes
You can use a bowl and hand mixer if you don't have a stand mixer. There's just a lot of mixing, so your arm will get a good workout.
The powdered sugar is optional. Without it, the marshmallows will just have more of a sticky or gelatinous feel rather than a soft and fluffy touch.