the hydrogen peroxide is what kills the bacteria (bacteria in your pits is the reason why they stank, not the sweat!!)
bi-carb then prevents future smells from occurring as it absorbs the smell
hydrogen peroxide is supposedly safe to use as a mouth wash... although i don't know enough about it to confirm that !!
i'm not claiming to be an expert by any means, but from what i understand its a very safe chemical.
https://www.youtube.com/watch?v=P0HqpVuMnP4Many Uses for Hydrogen Peroxide DYI
Combination of vinegar and hydrogen peroxide make a cheap, effective and non-toxic disinfectant agent and is said to be more effective at killing pathogens than bleach. As it is non-toxic, you can use it to disinfect fruits and vegetables, as well as pet toys, equipment and cages. In tests run at Virginia Polytechnic Institute and State University, pairing Vinegar and Hydrogen Peroxide mists, kills virtually all Salmonella, Shigella, or E. coli bacteria on heavily contaminated food and surfaces.