Bread substitute for wheat allergic
If you want to cast a wide net, searching for gluten-free bread might be your best here, even though is a wheat allergy rather than gluten intolerance, since removing the gluten necessarily means removing the wheat. Soy isn't too common an ingredient in breads so you should still be okay. 

And yes, you can make plenty of kinds of gluten-free bread at home, just search for recipes. Since gluten-free food is a big thing these days, a lot of people have put a lot of effort into figuring out recipes for gluten-free bread, so you should be able to find a variety. The tricky thing will then be finding the necessary flours. Bread normally depends quite a bit on gluten for structure, so it takes some effort to make recipes work well without it, and following them exactly is pretty important unless you're prepared for some serious trial and error. So when a recipe asks for white rice flour, brown rice flour, potato starch, tapioca starch, and powder, you want to either find those things or find a different recipe to try.



I don't know a lot about names of traditional wheat-free breads that you might be able to buy as-is, unfortunately, but I'm guessing that a lot of them will have a pretty different texture from wheat bread, so you might not be as happy with them. Of course, if you like Lithuanian black rye bread, you could certainly look for recipes for it or other rye-only breads.