Yes, natural materials can be used as hay covers, though they may have different effectiveness compared to synthetic options. Materials such as straw, hay, or burlap can provide a basic level of protection against rain and sun while allowing for some ventilation.
Still, while straw or hay can insulate and shed water, they're not as durable or effective as plastic or some fabric covers. Even tightly woven natural materials will tend to allow moisture to penetrate, and exposure to UV will inevitably cause degradation. It's also important to note that, while high quality geosynthetic materials can be reused from year to year, natural covers must be replaced every season.