I do think that it's part of the uniform, but for all we know their futuristic tight-fitting women's uniforms could also be a bra in itself.
Roddenberry did comment in The Making of Star Trek that women did not wear bras in Star Trek because they all had perfect breasts.
No, they just all take shore leave on the Baku planet, where the metaphasic radiation causes their boobs to firm up. Not that they care about such things in this day and age.
According to George Lucas and Carrie Fisher, there are no bras in space. Fisher said she hers were duck taped to hold them down. Ouch!
Only female Starfleet personel would wear bras. Men generally do not.