Is there any risk of illness from using baking soda or baking powder?

Probably not. Both are more likely to lose their potency as leavening agents after extended periods of time.  Exposure to moisture may prematurely compromise the performance of either. Baking powder generally includes corn starch to help keep active ingredients dry, but humidity and other sources of  ambient moisture may eventually take their toll. If you're not a  prolific baker, you should purchase baking powder in small quantities.  
 
Baking soda generally keeps longer than baking powder, though it also must be kept in a moisture-free environment. Some studies suggest baking soda may keep indefinitely under such conditions.
 
Both baking soda and baking powder are generally too dry to sustain the growth of  bacteria or mold. 
 
Source(s):
Ochef.com "The Potency of Baking Powder"
North Dakota State University "Food Safety Basics"
Survivalcenter.com "Food Storage FAQ: Cooking Staples"