Welcome to the forum.
Recently, purely out of curiosity, I did some research on chemical leavenings that can be used in a pizza dough in lieu of yeast. I was able to find a few recipes using baking powder, but the crust is not likely to be nearly as good as one using yeast because there isn't the same dough conditioning and flavor that come from using yeast. When chemical leavenings are used, they are usually a combination of baking soda (sodium bicarbonate--the Arm & Hammer stuff) and sodium aluminum phosphate--more commonly referred to as SALP. One brand of baking powder that apparently includes these two ingredients is the Calumet brand.
An example of a recipe that uses a combination of regular yeast and the abovementioned chemical leavening composition is the following one, from Tom Lehmann, of the American Institute of Baking, and a frequent contributor to the PMQ magazine and online forum: http://www.pmq.com/recipe/view_recipe.php?id=60
. The recipe uses baker's percents so some work will have to be done to convert the recipe to a smaller quantity of dough.
An improvement over the baking powder is a fairly new product called WRISE. It is put out by a company called The Wright Group (thewrightgroup.net) and is a fat coated-encapsulated leavening system made with the correct amount of soda (sodium bicarbonate) and SALP. Its use is apparently at the rate of 1% of weight of flour in a recipe like the Lehmann recipe where a small amount of regular yeast is also to be used, for the reasons mentioned above. If someone is allergic to yeast altogether, then the WRISE product can be used all by itself, at the rate of 2% of the weight of flour.
All of the above about WRISE is the good news. The bad news is that it is sold in only 50-lb. bags. However, today I called The Wright Group (1-800-201-3096) and asked if it was possible for a private individual, such as one with an allergy to yeast, to buy a smaller quantity of WRISE. The person I was referred to is Azarel Nieves, at 337-783-3096, x117. He was out of the lab today but it may be worth calling him if you are genuinely interested in trying out the WRISE product. If you are able to get the WRISE product, I may be able to help you downsize the Lehmann recipe to a small dough ball for a single-size pizza.
Another way of increasing the crispiness of a chemically-leavened dough, which is a quality you appear to want, is to add some vital wheat gluten to the dough ingredients. It is added dry to the other dry dough ingredient at a rate of 2-3% of the weight of flour.