Getting contact lenses in Japan - Megane Super
There are a lot of contact lens and eye glasses super stores around Tokyo, but I have always had a good experience at Megane Super.
You do need to get an eye exam the first time, but that is easy enough and not too expensive, probably about 3500 JPY.
Then they fit you for your lenses, most likely have them in stock unless you have strong astigmatism or something. Get your boxes right there, and then have your card made so you can just go back and buy the same ones again if you run out.
They also have good sales on solutions, etc.