Hunting: Supplies quests are one of three different hunting side missions, with this one requiring you to kill the required animals with a specific weapon, like a crossbow. You'll be given a certain area and have to find the animals in that area. Kill them, skin them to get the required number of meats and deliver those to a spot to finish the mission.
There's only four Hunting: Supplies missions in the game, all of which will be found by liberating outposts. No real substantial rewards are given out for doing them, except for Rupees bonuses for finishing two and all four missions.