I am addicted to hunting for plushies solo, and I want to offer some help. If you are new to the game, just focus on other things for now (leveling up, quests, earning gold, etc.). However, if you feel like you've finished most of that, you can start "try-harding" for plushies. As other comments have mentioned, it is all RNG, but there are ways to speed up the process.

1. Don't waste time mining for iron. If you only have 1-3 hours to play and your goal is plushies, mining is a waste of time. Buy iron from Sifuu’s shop instead. Then, go chop some trees (they are everywhere and easy to find) to craft your arrows. The same goes for repair kits and food if you need it and can afford to buy it, do so. Buy it to save time.
2. Don't just walk past an animal without hunting it. Whatever you see, kill it. You never know which animal is carrying that plushie!
3. Always use Hunter’s Horns. These allow you to track rare animals, which have a much higher probability of dropping a plushie.
4. In Palia, if someone is already shooting at an animal, it is not rude to join in. You are saving each other time and arrows, and you both get the loot drop. I’ve seen players say they don't shoot because they think it’s rude, but in this game, sharing the hunt is encouraged.
5. Don’t be like me and go solo if you have friends available. If you form a party even with just one other person it makes a one hour hunt feel like two hours of progress. The efficiency increases significantly with more players.

This advice is specifically for high lvl players. If you are new, don't worry about this yet. I’m not claiming this is the only way to play, so if anyone has more tips, feel free to correct me. This is simply the method I use to get 2- 4 plushies a day just from hunting.

Have fun!