You’ll obtain the Legendary Animals Map from Hosea during the story mission Exit Pursued by a Bruised Ego early in Chapter 2. The map will indicate the general location of these unique animals in the map, though finding them will require several steps.
First, you have to enter the legendary animal’s territory. You’ll get a prompt in the upper left corner of your screen if you’re within one. Once you’re inside the territory, the next part is finding your first clue. Legendary animals will always have up to 3-4 clues that you have to track and discover in order. Fortunately, the first clues are always found in the same spots, making it easier to track them. Use your Eagle Eye to find the yellow particles leading to a clue. Keep an eye out for a small question mark in your radar as well as it indicates the exact spot of the first clue. Once you examine the first clue, it’s recommended to apply the Cover Scent Lotion to hide your scent.
After finding and interacting with the first clue, it’s all about following the tracks and interacting with the following clues. Once Arthur says that the creature “must be near”, use your Eagle Eye and scan the surroundings. The tracks should lead to the legendary animal within shooting range. As long as it doesn’t smell you or you don’t spook it, you can attempt to get a bit closer to have clear shots. Most legendary animals will flee so you have to make sure your shots count.
Legendary animals will always drop their legendary skin/pelts and parts no matter what weapon you use. As such, it’s recommended to use your most powerful guns and rounds against legendary animals when hunting them, especially the most dangerous ones and the larger ones.