Camp (including cooking) equipment was carried in wagons on a regimental or company level as conditions allowed or stored during the campaign season. When a regiment went into winter camp, the equipment was returned or at least that is what was supposed to happen. Personal camp items for enlisted were not allowed on these wagons.

Some ranking officers may have had their own marked box which made it onto the wagons. Ideally when an item such as this is reproduced, the seller should provide you with some documentation as to the original. You could ask the person you bought it from.

As for food, it was carried in wagons, in bulk. An example would be barrels of molassas, flour and such. Even items such as salt, vinegar etc. were issued in bulk to companies or messes and then divided among the men. The food was then prepared and consumed or placed in the haversack for later.

An item called a "camp kitchen" would most likely have existed in very early war before the unit was mustered into service and became dependent on the respective commissary department and army transportation. It also may have been the property of an officer who had enough rank to justify his own food preparation equipment.

A good book to explain all this is "A TASTE FOR WAR, A Culinary History of the Blue and Gray" by William C. Davis.