Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
Place the whole oranges in a pot, cover with water, and boil for 1 hour until soft. Let them cool, then cut into quarters and remove any seeds.
Blend the oranges in a food processor until smooth. You should have about 1 cup of orange puree.
In a large mixing bowl, beat the eggs and sugar until light and fluffy, about 3 minutes.
Add the orange puree, vegetable oil, and vanilla extract to the egg mixture, mixing well to combine.
Sift together the flour, baking powder, baking soda, and salt. Gradually fold the dry ingredients into the wet mixture until just combined.
Pour the batter into the prepared cake pan and bake for 50-60 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before serving.