Preheat oven to 325 degrees F (165 degrees C). Grease 2 8×4-inch loaf pans.
Whisk eggs in a large bowl. Add vegetable oil and pumpkin puree; whisk until smooth.
Mix flour, sugar, vanilla pudding mix, baking soda, cinnamon, and salt in a separate bowl; stir into pumpkin mixture. Pour batter into the 2 prepared loaf pans.
Bake in the preheated oven until a toothpick inserted in the center of each loaf comes out clean, 75 to 80 minutes.