It's very seldom that a landing party really expects to get into a firefight, though.
And being armored against an unexpected attack isn't really worth it: if somebody really bothers to ambush a Starfleet landing party, they'll be sure to prepare well enough that the attack will
succeed, quite regardless of whether our heroes beam down in t-shirts or in a main battle tank. The true defense of a landing party is its ability to posthumously vaporize a continent... Nobody really dares attack even a naked Starfleet officer when knowing that this will cost him his entire civilization.
So that basically leaves just two scenarios where donning heavy armor makes any sense:
1) the landing party will engage in an infantry attack
2) the landing party will face savages who have no idea who they are up against - which means they will attack with things like spears, axes and sharpened avocados, as higher tech would imply knowledge of opposition
And the first scenario might never take place. Perhaps there exists no infantry action in the Trek universe? We never see it happen, after all: the closest we come is a desperate impromptu defense of an installation by a stranded starship crew against forces from a similarly downed enemy vessel, in "Siege of AR-558" and "The Ship".
Whether light body armor or forcefield belts/harnesses are a good idea in the second scenario... Difficult to say. The already discussed futuristic riot shields of ST5:TFF might be the more usual approach. But it seems that whenever our TOS heroes deliberately engage savage natives, they do so in the course of a mission where they try to hide their outer space origin and nature. Any protection would have to be of a concealable type, then, akin to the tiny Type 1 phasers.