I got this great question from a friend recently and figured I'd share my response.... to be honest I was really surprised to even see that this is a thing... because I've never experienced it with any of my clients and certainly not now that I'm a nursing momma myself!
The answer is a firm YES.
It is very safe to exercise while breastfeeding, and there appears to be no impact on milk supply.
There have been many studies done on breastfeeding and exercise and all have come back concluding that there is no significant effect from exercising on milk supply.
(This is just one study out of many... but I like it because the moms exercised at a moderate intensity 4x per week.)
If there is any take-away here, we can see how much better the moms felt during and after exercising. The moms improved their cardiovascular health, flexibility, musculoskelatal strength, and lost a few pounds.
Now, when I first got certified 10 years ago, I was forewarned by my instructor, Annette Lang, that chest-specific exercises could possibly alter the taste of momma's milk. The idea was that the lactic acid built up in the chest muscles would make the milk taste sour, and baby may reject it.
After 10 years of working with moms I have found this to be just a old wives tale... I've never had anyone note any problems after our workouts. I've also fed my son between sets of pushups with no issue.