Both baking soda and baking powder are leavening agents--they make cookies and cakes rise. Baking soda needs an acid to work, which is why you often find recipes using both baking soda and buttermilk. Baking powder is baking soda with the acid already added--and that acid is often cream of tartar, which is an acid. Cream of tartar by itself won't make the cookies rise.
Since baking soda is high in sodium, I tried googling low sodium gingerbread cookie recipes, but they all seem to call for at least 1/2 a teaspoon of baking soda.
There is a salt-free baking soda out there. It might not contain whatever ingredient your family is sensitive to--I think it is also gluten free and free from a couple of other things I can't remember off-hand. I realize this doesn't help you right now, but it's something to look into for the future.
You could try the cream of tartar. Or you could try your regular recipe without the baking soda/powder. The cookies might not come out the way you expect, but they might taste okay.
I did come across a gluten-free recipe that didn't have baking soda, but it required brown rice flour, which most people don't have in their kitchens regularly.