OP, Here's one route you can take if you want to avoid buying a bunch of products (gf cakes take a few different flours to get your flavor and texture profile right, so you'll have to buy a few things you may not end up using up if you go the from scratch route). Namaste foods offers a vanilla cake mix, chocolate cake mix, and spice cake mix (which is easily made into a carrot cake). http://www.namastefoods.com/products/cgi-bin/products.cgi?Category_Id=4
I tried their spice cake mix made as a carrot cake before I attempted my own scratch cakes so I could compare mine with products on the market. It was quite good - good texture, lovely flavor, but baked up fairly flat (probably because of the lack of baking powder - most baking powders have corn starch in them).
I'd go this route if you're not a gluten free baker - partly because it'll save you buying the myriad of ingredients, and partly because you'll avoid cross contamination (i.e. if you've ever scooped a tbsp of another of your ingredients, like sugar, etc. after having used the same spoon for flour - the sugar, etc. would be contaminated. I encourage you to do some research on how to avoid cross contamination as it is extremely important in keeping a gluten/corn/etc. intolerant person safe and healthy. I know I react very severely to even trace amounts of gluten. So, clean everything very well ahead of time, wipe down your countertops, all parts of your mixer, re-wash your cake pans and get into the corners, use silicone spatulas (not wooden spoons as they can hold gluten in the crevices)....a.k.a. - super-clean everything
There's lots of information online about baking safe allergy friendly foods - so research a little first so you keep your client safe.
It looks like Namaste also carries a frosting mix, but you can also make safe frosting just using crisco or palm oil, and subbing coconut milk or water for any milk. Do double check any flavorings (vanilla extract, etc.) to be sure they are safe.
I hope that helps!