1. Make cake: Preheat oven to 350ºF. Mist a 9-by-13-inch baking pan with cooking spray. Sift together flour, cocoa, baking powder and salt. Mix buttermilk, coloring and vanilla in a bowl.
2. In a large bowl, using an electric mixer on medium speed, beat sour cream, oil, butter and sugar until fluffy, about 3 minutes. Beat in eggs one at a time, scraping down sides of bowl. Add half of flour mixture, then buttermilk mixture, then remaining flour, mixing after each addition until just combined. Mix vinegar and baking soda until mixture fizzes; quickly fold into batter. Scrape batter into pan and tap on countertop to burst any air bubbles. Bake until a toothpick inserted in center of cake comes out clean, 30 to 35 minutes. Set on a wire rack to cool completely.
3. Make frosting: Using an electric mixer on medium-high speed, beat together cream cheese, yogurt, butter and vanilla until well combined. Slowly beat in sugar. Chill for 1 hour.
4. When cake is cool and frosting is chilled, spread frosting on top of cake. Keep refrigerated until ready to serve.