Kim - Here is one recipe previously posted here on Recipe Link. You will find more if you do a search for "chocolate mayonnaise cake" or "mayonnaise cake" Yummm
Kel

1937 Chocolate Mayonnaise Cake

Here's a Depression chocolate cake that substitues Hellmann's Mayonnaise for butter and milk. Note that Best Foods Creative Kitchen now gives "light" mayonnaise as an option for "regular". From American Century Cookbook: The Most Popular Recipes of the 20th Century.

2 cups flour*
2/3 cup unsweetened cocoa
1 1/4 teaspoons baking soda
1/4 teaspoon baking powder
3 eggs
1 2/3 cups sugar
1 teaspoon vanilla
1 cup Hellmann’s or Best Foods real or light or lowfat mayonnaise dressing
1 1/3 cups water

Preheat oven to 350 degrees F. Grease and flour bottom of 2 (9 x 1 1/2-inch) round cake pans.
In medium bowl, combine flour, cocoa, baking soda, and baking powder; set aside.

In large bowl with mixer at high speed, beat eggs, sugar and vanilla 3 minutes or until light and fluffy. Reduce speed to low; beat in mayonnaise until blended. Add flour mixture in 4 additions, alternately with water (begin and end with flour). Pour into prepared pans.

Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool in pans on wire racks 10 minutes. Remove; cool completely on racks. Frost as desired. Makes 2 (9-inch) layers.

*Recipe can be prepared with cake flour. Substitute 2 1/4 cups unsifted cake flour for flour.