This is Patricia's favorite easy and battle-tested cake recipe.
Preheat oven to 375. Mix dry ingredients together. Mix the oil, water, and vanilla extract in a separate bowl. Add the mixed wet ingredients to dry and stir until mixed. Add vinegar and stir. Put the cake in the oven immediately and bake for 25 to 30 minutes, or until a wooden toothpick inserted into the center removes cleanly.
For cupcakes, the cooking time needs to be decreased somewhat (bake for only 10-15 minutes. Check doneness often.)
As for frosting, I suggest dusting the cupcakes with powdered sugar, or using one of these readily available store-bought frostings:
It's easier than bothering with soy milk and vegan margarine or shortening that you may have to go to Whole Foods or a health food store to find. You could always substitute those for butter and milk in standard frosting recipes, if you are feeling ambitious.
There are other store-bought frostings that are dairy-free, but many frostings have milk-derived ingredients.