My son cannot eat anything that contains yeast. Are there any types of yeast-less bread that can be used as a sandwich bread?
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.
I'm sensitive to yeast also and I find Irish soda bread my answer. It is made with; 1lb plain flour. 1 tsp baking soda (soda bicarbonate), 1 tsp salt, 1tsp sugar. 11/2 - 2 cups buttermilk.
Sieve these ingredients together. Make a well in the middle, add the buttermilk and mix into a soft dough. Draw it all together into a circle about 1" thick, cut a cross in it, to aid cooking and cook on a flat cookie tin at 200 for 45 mins. When base sounds hollow on tapping it's cooked. Enjoy It's delicious. Helen
Here's a good-looking recipe: http://www.deliciousasitlooks.com/2013/03/gluten-free-low-fructose-irish-soda.html
Also, it seems the brand Kinnikinnick has a good Yeast Free Tapioca Bread that may be up your alley, according to this blogger: http://cook-aunaturel.blogspot.com/2009/03/yeast-free-gluten-free-portobello.html
First of all, "sourdough's" are yeast fermented breads, as has been stated above. If your son is allergic to all forms of yeast, stay away from this.
I'm Irish and my Mom makes Irish Soda bread all the time. I hate it. Won't eat it. But the rest of my family and plenty of other people like it and eat it. It's plenty firm and you can definitely make sandwiches from it. You can give it a try.
You could also try Salt Rising bread. It's a very regional bread and has a very unique flavor and smell. In its genuine form it's not yeast based. Don't buy it though unless you personally know the baker and can talk to them. Many "Salt Rising" breads available for sale are actually yeast based. Make it yourself or know your beaker. It's a very nice bread.