Place a rack in middle of oven; preheat to 350°. Coat 2 (9- x 1 1/2 -inch) round cake pans with cooking spray. Cover bottoms of pans with parchment paper. Coat parchment with cooking spray; set aside.
To make cake: Combine flour, sugar, cocoa, baking soda, and salt in a large bowl, whisking well. In a separate medium bowl, combine coffee, oil, egg whites, vinegar, and vanilla, whisking well. Combine coffee mixture and flour mixture, whisking until smooth. Divide batter evenly between the two prepared pans; tap pans on a work surface to break air pockets.
Bake until a wooden toothpick inserted in centers of cakes comes out clean (30-35 minutes). Cool in pans on a wire rack for 20 minutes. Run a knife around outside edges of cakes; invert cakes onto cooling rack. Remove parchment; cool completely on rack.
To make frosting: Place egg whites, sugar, cream of tartar, and salt in a large deep heatproof bowl set over about 1 1/2 inches of simmering water; beat with an electric mixer at medium-high speed until peaks form when beaters are lifted (about 7 minutes). Remove bowl from saucepan; continue beating until frosting is cool (about 3 minutes). Beat in vanilla.
Carefully place 1 cake layer on a plate. Using a spatula, spread a layer of frosting over top of cake; top with remaining cake. Spread remaining frosting over top and sides of entire cake.