Both my cooks have the Oak Campfire Kit recipe. As you've mentioned, the wiki doesn't say how to get it on the recipe page although the linked recipe index page says it's a "Basic Recipe" which implies it's automatically given upon reaching Eastemnet Cook tier. Since the recipe page isn't linked from any barterer page. I can only conclude at this point that the recipe is automatically given upon reaching the Eastemnet Cook tier.

https://lotro-wiki.com/index.php/Item:Oak_Campfire_Kit