We may be confusing a few related things which aren't exactly the same. Here's my penny's worth- I work in dermatology.
Fungal infection is possible but is unlikely to be both armpits at once (treat with creams from chemist, Selsun is an anti-fungal treatment).
Irritation from chemicals is unlikely as this would be just as true in a normal day as one spent kiting. (In my experience changing washing powder rarely helps skin problems as modern washing wachines are very good at rinsing residues, old style machines may have left some behind)
To my mind physical irritation ie rubbing is the culprit here. Looking at your clothes to ensure they aren't adding to the problem is worthwhile. You either need them very close- fitting ie rash vest, or loose fitting enough so that they stay free of the problem area. Talc is probably of benefit as long as you don't sweat a lot, it can cake with the moisture and add to the problem.
If you do find the irritation there, slap on lots of moisturising creams.