There's no particular reason that the UNSC couldn't have done it - with massively different physiologies, from each other and from humans, it's very likely that various species of Covenant could have been susceptible to agents that are harmless to humans. This would have forced them to wear hazmat/sealed gear on UNSC planets, possibly reducing the advantage of their shields. At least Elites, Jackals, and Grunts have demonstrated sealed-breathing systems in multiple games, and presumably the other members of the Covenant have their own equivalents.
However, humanity took very few prisoners, live or dead (the Covenant would glass any planet they didn't take with a ground assault), and fought mainly a defensive war. There weren't opportunities to attack Covenant civilians, just defend against the fleets and armies. Even if they got enough samples and made a breakthrough to create such an agent, in those situations, chem/bio weapons are of limited effectiveness against space-borne forces.
We saw, in fact, what would have happened if humanity had brought a bioweapon to a Covenant installation with noncombatants - the Flood overran High Charity in hours.
But, there's another reason that we never see the UNSC use these sorts of weapons, one that I think is more compelling: we as humans don't like writing about them. Chemical and biological weapons are so scary that we couch them in terms of Zombies, Vampires, Werewolves, and the like, because the reality is too terrifying. There really isn't a defense against them - witness mustard gas. Witness sarin. Witness white phosphorus. They're so hideously effective, and so very not testerone-fueled manliness, that who really wants to give them to the "good guys" in Halo?