Let's remember, body protection isn't around just
to protect against gunshots. Helmets on a modern battlefield are useful at protecting your head from shrapnel. On a more basic level, you can fall down, hit your head on a rock, and not crack your own skull open. Similarly, this would be a great argument for away teams to wear knee and elbow padding. Protective eyewear would be a must as well, seeing as lotsa UV can be very un-bueno for the ol' mk 1 eyeball. If I were in Starfleet, I'd probably vouch for better footwear as well. What they've got is ok for a day at the office, but its got an awful tread, and you'd get all kinds of soil, foreign stuff, and moisture over the top cause they're so short.
It's not just fights. Starfleet's really not so well equipped for a long-term stay in any environment that isn't SoCal Temperate. If I don't slip on a rock and give myself a concussion, I'm going to trudge through mud for a few hours and get trenchfoot. Joy of joys.
Effectiveness of armored vests/body pieces against phasers and disruptors is debatable... made complicated by the fact that we really haven't seen such armor in use by Starfleet. But the Federation has encountered people who are resistant to small arms, either through biology, cybernetic augmentation, or external gear. (The Hunter guys from DS9, the Borg, there's more, but I don't quite recall). I'll bet that the Federation can reverse-engineer that kind of stuff, eventually.
However, even if body armor is infeasible at the moment, I can't see Starfleet not
wanting to protect its people in a fight. Maybe little deployable theater shields or something?