In the story one of the main missions (at around ~30%) demands you to get meat, meat is only available through hunting. So you won't be able to get through the story.
Around the same time there is a side mission where you are asked to create a bag, which is made from animal hide, again obtained through hunting.
Another side mission demands that you get a cooked fish, which you need to fish for, but this is a side mission, not main mission.
Also, there are enemies which are like crabs that attack you (in the story it's after the meat mission). Don't know if that counts as animals, but they are available in both the story and the challenges, so if it counts, you won't be able to do the challenges as well as they will kill you.
Later on in the story and in the challenges there are "zombie deers" and "zombie ostriches" which attack you. Again, not sure if this counts as animals or not.
There might be more missions where animal parts are needed, I haven't finished the game yet.
Best Answer
Just played through the game twice. I could only find a pulley Santa Barbara. It is the last/largest island.