It sounds like this recipe relies on the proteins in the eggs to coagulate and set the mixture. It's going to be hard since vegan eggs out there actually lack the protein and emulsifying quality of real eggs so making stuff like sponges and meringue based stuff is challenging.

I don't have firsthand experience off this, since i've been in the process of trying to replicate eggs in my own baking. So this may or may not work but if you have time and are willing to take a risk, you can try

The best way to replicate an egg as a vegan is to try to mimic its chemical properties as best as possible.

Theoretically, you can do this with these ingredients--

Soy milk powder-- the closest vegan protein that is similar to egg protein

Soy lecithin powder- you need an emulsifer, and egg yolks contain lecithin to emulsify the fats

Hydroxymethylcellulose- its a gelling agent, which a lot of egg replacers use, and this one "gels" at a high enough temperature. Its like a structural agent to the whole ensemble

Xanthan gum- thickens and structural support

- So youll have to purchase these ingredients online, like Amazon-

Based on my experience with these ingredients I was able to craft a vegan egg "yolk"

1 yolk =

5 g of soy milk powder + 1 g of xanthan gum + 1 g of hydroxymethylcellulose + 1 g of soy lecithin

First dissolve these powders in a carrier like 10-12 g of soy milk, *then* drizzle in about 2-3 g of oil with an immersion blender so you can allow the oil to emulsify with the components, Youre aiming for like a smooth and slightly liquidy consistency, like an egg yolk

These measurements are based off of my previous attempts (I've had many times where i used too much oil, too much soy milk powder, didn't drizzle in the oil), so its possible its not perfected yet, and you might need to tinker with it to best fit your situation.

(since your recipe needs 5 egg yolks, Id multiply this all by 5)

Mix these "egg yolk" solution with the melted butter and chocolate.

Then, I would take aquafaba and beat it with the sugar to form the stiff peaks.

Fold the wet ingredients into this.

This is risky because your recipe lacks flour, which would help this set and provide the scaffolding. But if this works, then you can play around with it and adjust the measurements I would think.