I just made a recipe tonight that would work for you! It is from the Betty Crocker Classic Cookbook- no eggs or dairy in it. You can mix by hand as well. It turned out well and was a bit denser than traditional recipes, but not in a bad way.
1 1/2 cups all-purpose flour
1 cup sugar
1/4cup baking cocoa
1 teaspoon baking soda
1 teaspoon salt
1/3 cup vegetable oil
1 teaspoon white vinegar
1/2 teaspoon vanilla extract
1 cup cold water
1. Heat oven to 350°. Grease bottom and side of round pan, 9 X 1 1/2 inches, or square pan, 8 X 8 X 2 inches, with shortening; lightly flour.
2. Mix flour, sugar, cocoa, baking soda and salt in medium bowl. Mix oil, vinegar and vanilla. Vigorously stir oil mixture and water into flour mixture about 1 minute or until well blended. Immediately pour into pan.
3. Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool 15 minutes. Serve warm or cool with ice cream.
*Do not use self-rising flour.
NUTRITION FACTS: 1 Serving:; Calories 230 (Calories from Fat 70); Fat 8g (Saturated 1g); Cholesterol 0mg; Sodium 270mg; Carbohydrate 38g (Dietary Fiber 1g); Protein 2g % DAILY VALUE: Vitamin A 0%; Vitamin C 0%; Calcium 0%; Iron 4% DIET EXCHANGES: 1 Starch; 1 1/2 Fruit; 1 Fat
Also, look here:
http://www.cookiemadness.net/?s=vegan
Anna has some vegan recipes for cookies/cake, although I've never tried any. Another idea is 101cookbooks.com for possible vegan baking b/c Heidi's recipes are veggie ones only, but I think she has done a lot of vegan stuff too w/ her baking. HTH!