I'm a little bored writing "read the walkthrough" every time there is a question here but I guess that's the way it is.
To get the membership card you need to take the thing that looks like a pickaxe but is called "manhole opener" in the building next to the gymnasium in the school area, in the third town. Then go to the second town and use it on the manhole in the water space surrounded by trees to get to the underground area. Go right then up to get to the ladder then go left and take the "Nudist island membership card" you saw from the ground.
I hope that helped you
