To unlock Emiko you need to be in year two and in the summer season of your game. Leave your farm on a non-rainy day between 9:00 pm and 2:00 am, and you will trigger a cutscene.
In the cutscene your character discovers something strange about the waterfall. As you go to investigate it, you find a path leading through it. Your character will automatically go down the path for the first time, where you will meet Emiko.
After that cutscene you can visit her whenever you want after that, with the exception of rainy or stormy days (you can enter the waterfall, but you cannot enter her shrine/house). She is also not avaiable after 8:00 PM (though she may be outside the shrine).