Preheat oven to 350 degrees F (175 degrees C). Grease and line 4 8-inch round cake pans with parchment paper.
Combine white cake mix, egg whites, vegetable oil, milk, and vanilla extract in a large mixing bowl. Beat for 30 seconds using an electric mixer on low speed. Increase mixer speed to medium and beat for 2 more minutes. The batter will still be slightly lumpy.
To make the red cakes, measure half the batter into a separate bowl and whisk in red food coloring. Evenly transfer the red batter into 2 of the 8-inch prepared cake pans; smooth tops of batter with a spatula. Transfer half of the remaining uncolored batter into a 3rd prepared cake pan; smooth top as before. This will be the white cake portion.
Add blue food coloring to the remaining uncolored batter in the large mixing bowl; fold 1 cup white chocolate chips into blue batter until combined. Transfer batter to remaining prepared 8-inch cake pan; smooth top of batter with a spatula.
Bake all 4 cakes in the preheated oven until a toothpick inserted into the center of each cake comes out clean, about 30 minutes. Remove the cakes from oven and place onto cooling racks until cooled completely.
