How do Crispbread manufacturers use so few ingredients?

All crisp-bread recipes use water (or at least some ingredient with a high water content), but the bread is baked and then hung from the ceiling to dry further until the water content is very much reduced.

The real mystery is how structure is introduced without leavening. Wikipedia has good information on this point,

[...] bubbles are introduced into the dough mechanically. Traditionally, this was done by mixing snow or powdered ice into the dough, which then evaporated during baking. Today, the dough, which must contain a large amount of water, is cooled and mixed until bubbly.

Another method is to knead the dough under pressure in an extruder. The sudden drop in pressure then causes water to evaporate, creating bubbles in the dough.

Presumably knäckebröd was only made during the winter months before its production became industrialised. For the modern home cook, the difficulty of introducing structure still remains, so it is unsurprising that you didn't find many recipes with very basic ingredients.

Some recipes for home cooks introduce structure in the form of seeds and whole grains. Others, as you noted use yeast.

Some Swedish crisp breads such as tunnbröd, have no leavening, no gas incorporated into the dough and are wafer thin. These can be made at home with a very short ingredient list, no specialised equipment and during any season :-)

At least some of the major Swedish manufacturers do use yeast. My favorite, Leksands knäcke, for instance uses,

swedish wholegrain rye flour, fresh water from our own spring, yeast and a pinch of salt.

So if you are considering baking your own knäcke, you don't necessarily have to avoid recipes using yeast.