Temperate Rainforests
Yes, the Appalachian Region would be considered a Temperate Rainforest based on a combination of rainfall numbers, humidity and temperatures. Its latitude runs from in the 30's and up to high 50's in latitude location north of the equator. Feeding the forest region are a combination of two damp air masses which rotate in a cyclonic rotation inward from the marine locations, namely the Maritime Tropical air mass bringing very humid air from the Gulf of Mexico and the cooler air but also damp Martime Polar air mass. The mountains, while lower in elevation than the Western Rocky Mountains, offer much of this damp air from either air mass to move over toward the mountains when the jetstream is positioned in a way to not influence these air masses in a way that pushes them eastward. Orographic type precipitation can occur as a result, but much of it in that region is frontal as the colder Contintental Polar and Arctic air masses tend to interact with the damp air masses mentioned previously dumping significant amounts of rainfall suddenly as I can attest in my many journeys through these mountains on my way to Georgia or Florida from Ontario. Given the rainfall alone and the fact there is a diverse complexity of vegetation and animal specie, I would surely believe that the Appalachian Region qualifies easily as a Temperate Rainforest. Climate graph > location in Appalachian Region https://en.climate-data.org/north-america/united-states-of-america/virginia/appalachia-135846/#climate-table  and, location map of North American air masses paths running along and to this region https://wordpress.accuweather.com/wp-content/uploads/2018/01/static-winter-air-masses-11-am-1.jpg?w=632