Preheat the oven to 350 degrees (fahrenheit). Butter and flour a 9-inch springform cake pan.
In a small bowl, sift together the flour, baking powder and salt. In a larger bowl, beat the eggs until thick and lemony. Whisk in the sugar, yogurt, lemon zest and vanilla.
Slowly pour the dry ingredients into the wet, stirring constantly with a wooden spoon, until thoroughly combined.
Using a rubber spatula, fold the vegetable oil into the batter, making sure it’s fully incorporated.
Fold in the cranberries and pour the batter into the prepared pan.
Bake on the middle rack for about 50 minutes, until a tester inserted in the middle of the cake comes out clean.
While the cake is baking, prepare the syrup: combine sugar and lemon juice in a small pot and heat over medium heat, stirring until the sugar is dissolved. Alternatively, combine sugar and lemon juice in a mixing cup and microwave at high for one minute. Set aside.
Remove the cake from the oven and allow to cool in the pan, on a rack, for 10 minutes. Run a slim blade around the edge of the cake and remove the rim.
Place the cake on a plate and pour the syrup slowly over top, allowing it to soak in. Let the cake cool to room temperature.
Sift the icing sugar over top of the cake, just before serving. Serve as is or accompanied by a dollop of whipped cream.