If you look, you can find a variety of eggless cakes that involve oil, bicarbonate soda and vinegar. This recipe started there, but, well, it's something slightly different now. So, here goes.
You will need:
1 ⅔ cups self-raising flour
1 ½ tsp allspice
1 cup sugar (it called for brown sugar, but we only have raw)
½ tsp salt
½ cup water
½ cup applesauce
⅓ cup oil
I used SR flour, so didn't need the bicarbonate soda and vinegar.
And this is what you do with it:
First, you DO NOT grease the pan. Or maybe you should, but the person who wrote the recipe said DO NOT (just like that) so I'm just passing it on. I didn't this time, but I think, maybe, I will next time.
The whole point of the recipe is that it is incredibly easy, and doesn't involve much in the way of clean-up, so you're supposed to mix everything in the pan you're baking in. So, dump all the dry ingredients in your 20cm (8") square pan and whisk until blended. Then you add the wet ingredients, and stir to mix, not overbeat.
Bake at 175C (350F or thereabouts) for 35-40 minutes. I baked it in my Mum's oven at 180C for 30 minutes
because her's doesn't have clear markings. Also, her oven is fan-forced, so I probably should have set it lower. Anyway, it was certainly edible. I made it a couple of hours ago, and it's mostly gone now.
This is the Applesauce cake. You may even be able to read a couple of recipes off my sheet. I laminate my recipe sheets so they actually survive my cooking :o

This is the chocolate cake. I've put the recipe at the bottom of this entry. Hmm... Let me see. Milk, oil, salt and vanilla. Some of the ingredients.

If you are using a special cake pan, it might be a good idea to mix it in something else, and pour it into the pan, so you don't get unmixed flour in the grooves.

Mum gave this to me a couple of Christmases ago. It's bigger than the 20 x 20cm pan, so it cooked quicker - about 24mins. Nice, hey.

The chocolate cake recipe, using the same method, is this:
1 ½ cups self-raising flour
1 cup sugar
⅓ cocoa
½ tsp salt
1 cup water (I used milk)
½ tsp vanilla
⅓ cup oil
So, again, whisk together the dry ingredients. Add the wet, and mix. My suggestion would be to do this in a separate container, and pour into a lined 20cm (8") square pan, because both my cakes (in square pans) stuck. I may just have to stick the chocolate cake back together with icing. Sigh... the things I have to do... ;)