Preheat oven to 325 degrees F (165 degrees C). Grease and flour a 9x13-inch pan.
Mix flour, baking soda, and salt together in a large bowl. Beat margarine and white sugar together in a separate bowl until light and fluffy. Beat in eggs, one at a time, until fully incorporated; stir in 2 teaspoons vanilla extract. Stir butter mixture into flour mixture, alternating with eggnog, until well combined. Fold in bananas. Pour batter into prepared pan.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 1 hour. Remove from oven and cool on a wire rack, about 1 hour.
Beat butter and cream cheese together in a bowl until creamy and smooth; beat in 1 teaspoon vanilla extract. Add confectioners' sugar; beat on medium-low speed with an electric hand mixer until combined. Spread frosting onto cooled cake.
