Combine the rhubarb, sugar, and water in a saucepan over medium heat. Bring to a bare simmer. Cook for about 25 minutes, stirring occasionally, until the rhubarb pretty much falls apart. Turn off the heat, add in the vanilla, and cool to room temperature before beginning to make the cake.
Preheat your oven to 350 degrees Fahrenheit and lightly grease and flour a nine-inch round cake pan.
Whisk together the non-dairy milk, orange juice, sugar, oil, and vanilla until thoroughly combined. Sift in the flour, baking powder, baking soda, and salt all at once, using a wide spatula to fold in the dry ingredients with as few strokes as possible. Being careful not to over-mix, stir the batter until just combined to achieve a tender crumb.
Pour half of the batter into your prepared pan, using a spatula to smooth it out into an even layer. Top with the rhubarb mixture, and then top with remaining cake batter. Swirl a knife through all of the layers to give it a somewhat marbled effect.
Bake for 40-45 minutes, tenting the top with aluminum foil after 30 minutes if it seems to be getting too dark. The sides should slightly pull away from the pan and the cake will be golden brown when done. Cool completely in the pan before serving.