Living a zero waste, no plastic life is a choice we have made to help live a greener more earth friendly way. That said, doing it with pets and livestock takes more creativity than usual. The horse pictured below is mine. He is an American Cream Draft cross named Butterfinger.
It's difficult to get feed in bags that aren't made with plastic. I do recycle those bags when I have to purchase them. I make them into farm bags to use at the grocery store, farmers market, hauling items to donate and where ever else I can. They do make them with paper but not always in the kind of feed you need. Buying in bulk is out of some folks price range but that is usually the best option.
Finding non GMO and organic feed is another concern. Growing for your animals is a great option if you have some land. It doesn't take a lot of room to grow a few herbs, some grass and carrots for rabbits or have a fruit tree. Rabbits can chew on the branches you take them and horses love apples. Please do your research on what varieties to get that won't harm your animals. Rise and Shine Rabbitry is a great place to start. Here is a link to a list of Medicinal Herbs for Rabbits.