Do you know what types of allergies? Wheat, nuts, etc.? I don't understand the vegan part, unless the child is allergic to honey, which would be horrible.
Anyway, maybe the Organic Bliss Deli and Bakery, located in Sylvania, could offer something. I visited this place when it was located on Secor Rd. I think it's the same place. They offer gluten-free food items, including many bread and sweet-treat options, assuming the child has a wheat allergy. From their site:
Gluten Free Bakery Menu - pdf file
And I would suggest visiting the Phoenix Earth Food Co-op and Health Foods by Claudia's to purchase products to prepare at home.
The food co-op's products page lists what the store sells, including some of their gluten-free items. And the people who work at both stores should be able to answer questions.