You could try Francis Simun Bakery in Dallas:
http://www.francissimunbakery.com/
They carry baked goods made with alternative flours, no dairy, etc. They're expensive, but they will custom make what you want with ingredients that'll work for your son's diet. I used to get some baked goods there several years ago. Two of my children had bad allergies before they took a treatment that improved them. We went to a doctor in Dallas at the time, and it was nice for them to have a bakery to just walk into and order something, because they couldn't do that anywhere else.
However, it's not that hard to bake a cake yourself with an alternative flour. Try spelt, because its taste is similar to wheat flour. They carry it at Whole Foods and other health food stores. Look online for recipes that are specifically adapted for wheat/dairy allergies. You might consider making something more like a muffin or banana bread sort of cake, since you can't use eggs. Francis Simun carries some muffins that don't have the ingredients that your son needs to avoid.
Now, after saying all of that, perhaps you might want to reconsider whether all of this trouble is worth it for the first birthday. Most babies end up smashing up their cupcake or piece of cake, and it makes a cute picture. He won't remember it, no matter what you do, except for the cute picture. You could make a cake for the family, and give your son a special treat of his own. If you handle it wisely, he won't even know he's missing anything.
I used to keep specially made cupcakes for my children in the freezer. When they would go to a birthday party, they'd have their own cupcake while the others had birthday cake.
One more idea: Make a big cookie instead of a cake, using spelt. My youngest son doesn't like cake (weird, isn't it?), so he always has a huge cookie instead of a birthday cake. The, you can easily avoid dairy and eggs.
Enjoy your son's first birthday!!
