"雷鸟科的猎鸟" can be translated as "hunting birds of the Tetraonidae family" in English.
Here, “雷鸟科” refers to the Tetraonidae family, which includes grouse and related birds, and “猎鸟” is translated as "hunting birds," indicating birds that are hunted. If the context implies that these birds are themselves the hunters (though this is less common for the Tetraonidae family), a more specific description like "predatory birds of the Tetraonidae family" (though note that Tetraonidae are not typically predatory) or a rephrasing to clarify the role might be necessary. But based on the straightforward literal translation, "hunting birds of the Tetraonidae family" is appropriate.