There are roughly four types of bread by leavening: yeast breads, chemically leavened breads, sourdough breads and unleavened breads.
Yeast breads are obviously out of question. Chemically leavened breads are made with baking soda or baking powder, and are also often called "quickbreads". They are less common as pure breads, and mostly available as special recipes which are somewhere between cake and bread, such as banana bread or zucchini bread. I don't know if you can buy any of them easily, but they are extremely easy to make at home (about as easy as muffins).
Edit per the commenters' suggestion, this is probably your best solution. While most recipes for quickbreads in use today are indeed for banana bread and such, there are older recipes (especially from wartime) where normal, non-flavored bread is made with soda. Using them is probably the best you can do. Caveat: the more breadlike you want the texture to be, the more dense will you have to make the bread, because developing the texture takes time, and baking soda and baking powder lose their leavening properties with time.
I would advise you to stay away from sourdough breads. They don't contain commercial yeast, but during fermentation they catch wild yeasts. The wild yeasts are closely related to commercial yeast, and likely to trigger the same allergies as commercial yeast does.
The last variety, unleavened breads, are flatbreads. They are dense due to lack of leavening, so if they were made thick, the center would not bake. This means that you can't cut them into slices to make a traditional sandwich. But if you relax your definition of what a sandwich should be like, you can use unleavened breads like tortillas or matzo layered with the ingredients, or make actual wraps instead of sandwiches. Pay attention though, not every flatbread is unleavened, naan for example contains yeast. Knäckebröd is a flatbread which can be leavened or unleavened depending on brand, and it makes excellent sandwiches.
You can also try making sandwiches with non-bread substances. A jibarito is a sandwich where the filling is packed in fried plantain slices instead in bread. Other vegetable slices like zucchini or oven-dried eggplant should work too. Puffed rice wafers are also a good sandwich packaging, as well as savoury waffles (make your own or read the packaging, because there are yeast-leavened waffles).
Other types of leavening are not used in breads (such as meringue leavening which is used for cakes only), but if you like baking and want to make your son's diet more varied, you can try adapting cake recipes to savoury, breadlike results. Adapting recipes is more for advanced bakers though, because the sugar in cakes plays important roles besides giving them flavor.
It is important to note that most of my suggestions are pretty different from the everyday yeast breads most people use for sandwiches. You probably won't get anything too similar to them (quickbreads will be the closest). Don't try to imitate them as close as possible, you will be frustrated. Teach your son (and why not yourself?) to appreciate the substitutes which make for very tasty non-traditional sandwiches with their own charm.
Best Answer
There are various types of bread without yeast. Unleavened Jewish Passover bread, for one, and soda bread for another. Unleavened bread doesn't have the spongy texture.
Basically, yeast has two uses in baking. 1. The spongelike structure and 2. Taste.