Preheat the oven to 350°F (175°C). Line a standard 12-cup muffin tin with paper liners.
In a large mixing bowl, whisk together the sugar, applesauce, and oil until smooth. Add the oat milk and vanilla, whisk to combine. Stir in the lemon juice and zest.
In a separate medium bowl, whisk together the flour, baking powder, baking soda, salt, and poppy seeds until evenly mixed.
Add the dry mixture to the wet mixture and stir gently until just combined. Avoid overmixing to keep the muffins tender.
Divide the batter evenly among the muffin cups (about 1/3 cup per cup). Bake for 18–20 minutes, or until a toothpick inserted in the center comes out clean.
Let muffins cool in the pan for 5 minutes, then transfer to a wire rack to cool completely before glazing.
In a small bowl, whisk the powdered sugar with 1 tablespoon lemon juice, adding more juice a little at a time until you reach a pourable consistency.
Drizzle or spoon glaze over cooled muffins. Let set for a few minutes before serving. Best enjoyed the same day.