Most any smell is from an organic compound. A good enzyme based natural cleaner would be the way to go as it will destroy the compound from sweat, body oils or dander and not harm the rubber or plastic parts.
I am not sure whether you are dealing with an acid or base chemically. Baking soda is a base and as such will neutralize an acid, vinegar is an acid and neutralize bases. So depending on whether you are dealing with a substance that is an acid or a base. Then it does make a difference whether you treat with an acid (vinegar) or a base ( baking soda) The reason why putting vinegar and baking soda together gives such a big reaction is you are mixing an acid and a base and get a vigorous reaction.
That would be my suggestion.