Put oven rack in middle position and heat oven to 375°F/190°C. Coat a 9x13 cake pan with margarine (I spray it with Pam).
Whisk together flour, baking powder, lemon zest, and salt in a bowl.
Beat together oil, mashed avocado, and granulated sugar in another bowl with an electric mixer at medium speed until pale and fluffy, 2 to 3 minutes. Add coconut milk and lemon juice. Reduce speed to low, then add flour mixture and poppy seeds and mix until just combined.
Transfer batter to the cake pan, smoothing top, and bake until a pick inserted in the center of the cake comes out clean and the top is golden brown, about 50 minutes. Let the cake cool.
For the Lemon Icing
Whisk together powdered sugar and lemon juice in a bowl until smooth. Add coconut oil and stir until combined. Add plant-based milk, one tablespoon at a time, until you get a pourable consistency. Pour over warm cake, spreading it with a spatula to drizzle over edges. Let stand until glaze is set, about 15 minutes.
For the Blueberry Sauce
Combine syrup, cornstarch, and water in a saucepan. Stir to combine. Add blueberries. Turn the heat to medium. Bring the mixture to a boil. Then remove from heat and gently stir in lemon juice and peel.