Do chickens lay year round

Yes, they do -- if the number of light hours is enough. Place lights in the hen house to elongate the laying season of hens. They will lay less often in the winter.

Of course, in the northern areas hens will not lay many eggs in the winter unless artificial light is provided, as the number of light hours is greatly decreased. We live in the Deep South, and our one lone chicken is still laying eggs in December/January, but not as frequently as she did during the summer.