Whisk together the dairy-free milk and apple cider vinegar in a small bowl. Set aside for a few minutes to curdle and form vegan buttermilk.
In a large bowl, combine the vegan butter, granulated sugar, olive oil, vanilla extract, vegan yogurt, and the prepared vegan buttermilk. Whisk until smooth.
Add the flour, cocoa powder, baking powder, baking soda, and salt to the wet mixture. Whisk everything together until just combined.
Stir in the red food gel or dye until the batter reaches your desired red velvet hue.
Preheat the oven to 350°F (175°C). Line a cake pan with parchment paper and grease it with oil. Pour the batter into the prepared pan and bake for 50-55 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, whisk together the vegan cream cheese, vegan butter, powdered sugar, vanilla bean paste or extract, and a pinch of salt until smooth and creamy.
Allow the cake to fully cool before spreading the frosting on top. Slice and enjoy!