If I know the name of the corporation from the recruitments page, how do I then look up that corporation in game to apply to it?
under people and places, you can search for either the person who posted the recruitment, or the corp itself.
There’s also a small search prompt top left in the GUI, there you can search for players, corps, structures, systems in EVE.
Some external info:
(note: you can also use it in the in-game notepad)