You’re really going to have to define “plastics” to get a good answer to that.
Plastic in the material science definition means any material that can be permanently deformed without breaking. So, lots of materials created by living things meet that definition.
If you mean thermoplastics, which is the more common colloquial definition, well, several things meet that definition as well, including horn and many other types of keratin.
If you mean polymerized hydrocarbon based thermoplastics, which is what you probably are thinking of, chitin is the most common answer.