Adjust oven rack to middle position and preheat oven to 400°F (200°C). Line a 12-cup muffin tin with paper or foil liners.
In a large bowl, whisk together flour, sugar, baking powder, baking soda, and salt.
In a microwave safe bowl, combine water, cocoa, chocolate, oil, and vanilla and microwave in 30 second intervals at 50 percent power, whisking occasionally, until melted and smooth, about 2 minutes. Alternately, in a saucepan set over low heat stir together the water, cocoa, chocolate, oil, and vanilla until melted and smooth, taking care not to burn the chocolate. Set aside to cool slightly.
Meanwhile, using a stand mixer fitted with whisk, whip aquafaba and cream of tartar on high speed until the mixture forms soft peaks, 3 to 9 minutes. Using a rubber spatula, stir chocolate mixture into flour mixture until batter is thoroughly combined and smooth, about 1 minute. Stir one-third of whipped aquafaba into batter to lighten, then gently fold in remaining aquafaba until no white streaks remain.
Divide batter evenly among prepared muffin cups. Bake until tops are set and spring back when pressed lightly, 16 to 20 minutes, rotating muffin tin halfway through baking.
Let cupcakes cool in muffin tin for 10 minutes, then move to wire rack and let cool completely, about 1 hour.
While the cupcakes are cooling, finish preparing the frosting. Using a stand mixer fitted with whisk attachment, whip at high speed until fluffy, mousse-like soft peaks form, 2 to 4 minutes, scraping down bowl halfway through whipping. Spread frosting evenly over cupcakes and serve.