It’s amazing what Allrecipes can do for you. I altered the recipe by Ginger P (Eggless Chocolate Cake II) slightly (as seen by the bolded ingredients below) and got THiS Beautiful piece of work 
Baked it as a birthday gift for my 6 year old cousin.
INGREDIENTS
5 cups of sifted all-purpose flour
2 1/2 cups of white sugar
1 cup of sweetened cocoa powder (president choice)
1 cup vegetable oil
2 cups water
1 cup milk
3 tsp baking soda
1/2 tsp salt
3 tbsp vanilla extract
INSTRUCTIONS
1. preheat oven to 350 degrees F (175 degrees C)
2. in a large bowl, sift together flour, cocoa powder, soda and salt. Add sugar and mix together.
3. Add oil, water, milk and vanilla and mix as thorough as possible
4. Pour into pan. Bake at 350 degrees F (175 degrees C) for 1 to 1.5 hour (all dependent on your oven) But KEY: rises nicely and toothpick inserted in the center come out clean.
5. I turned off the oven and left the cake to sit in it for a good 3 hours before taking it out of the oven to let it cool
(SN: I used a 9 x 13 inch non stick pan (no grease/ oil necessary)
What made this recipe so amazing isn’t the taste….cakes with eggs still taste better in general…but the fact that it rose so nicely even without eggs.
In terms of the frosting…you can easily find a good tutorial on youtube.
i will try this…. looks good!!