Preheat the oven to 350°F. Line a loaf pan with parchment paper, allowing some to hang over the sides, and coat the pan with nonstick cooking spray.
Add the flour, baking soda, and sea salt to a large bowl and whisk gently to combine.
Add the sugar, oil, milk, vinegar, lemon juice, and vanilla extract to the bowl, and whisk to combine. Be careful not to overmix; it's okay if the batter is not completely smooth, but you should not see any traces of flour.
Fold in the poppy seeds and lemon zest with a silicone spatula. Pour the batter into the prepared pan.
Bake the cake for about 42-47 minutes, until it is golden brown and a toothpick comes out clean. Let the cake rest in the pan for about 30 minutes, then carefully remove it from the pan and peel off the parchment paper. Place the cake on a wire rack over a baking pan to allow it to fully cool before you ice it.
Make the icing: whisk together the powdered sugar, non-dairy milk, and lemon juice. It should be thick but pourable. If needed, add another teaspoon of non-dairy milk. If needed, pour the icing through a mesh strainer to eliminate any lumps. Once the cake has cooled completely, drizzle the icing over the cake.