Whisk the sugar, cocoa powder, cornstarch, and salt together in a medium saucepan so there are no lumps.
Gradually whisk in the coconut milk, then the almond milk, until the mixture is completely smooth.
Cook over medium heat, whisking constantly, for 6 to 8 minutes until the pudding thickens and begins to bubble. Let it bubble gently for 1 full minute while whisking so the cornstarch fully activates.
Remove from the heat and stir in the chopped dairy free chocolate and vanilla until melted and glossy.
Divide the pudding into serving cups and press a small piece of parchment paper directly onto the surface of each to prevent a skin. Chill for at least 3 hours until silky and set.