Seconding the idea to use a mix, first-off. It's a birthday cake for a baby, you don't need anything fancy. My son had dairy-free Pamela's brownie mix for his first birthday cupcakes. The mix is GF, dairy-free, egg-free, though the chocolate chips are listed as "may have traces of dairy from processing."

If soy is OK, I've had good luck substituting Earth Balance margarine for butter, and 1/4 of a block of silken tofu (the kind that comes vacuum packed in the little boxes, not the kind that comes in water) for an egg in vegan baking.