Huh? Not allowed to wear a bra? That certainly wouldn't be an enforceable work rule in the US. Certainly laboratories and manufacturing floors etc may disallow shorts or dresses for safety or clean room reasons safety should always dictate that long hair be tied down or covered around machinery, again for safety, likewise jewelry may be unsafe in some work environments.
Anyhow back when I was regularly running longer distances I envied women the use of sports bras. That was before I was taking hormones but I soon learned that cotton could cause nipple chafe in even a couple of miles and the lowest friction gear would still likely involve some nipple bleeding as I got up over 10 miles.
If you can't argue the point of wearing a bra, then changing into a running shirt for exercise would help as would band-aids or similar over nipples -- lots of marathon runners use these.