It's likely just a modesty issue. We did have coaches require a bra at meets because their competition leotard had very thin fabric that showed everything (and that was for all levels down to the littles, probably as young as 7). Many of the girls started wearing them for practice after that as well. Consider that seeing the details of breast development could make coaches uncomfortable, and could also lead to teasing or even bullying amongst the kids. Also, it is sometimes easier for a gym to require everyone to do something to solve a problem rather than call out particular kids on a sensitive topic.
There are lots of cute and comfy sports bras available these days. I'd suggest explaining the requirement to your daughter and taking her on a trip to Target and Kohls to find some cute and comfortable ones that she likes and is willing to wear. If the band creates a sensory issue, another suggestion is a nude toned bodysuit liner as used in ballet, we had several girls wear these to address the modesty issue instead of bras, but caution that gymnastics and ballet leotards are cut differently so it can be hard to find one that won't show.