Harajuku is arguably one of the most famous fashion districts in the entire world. From the quirks of Takeshita street to the high end stores of Omotesando there isn't anything you can't find in this part of Tokyo. And that of course includes sneakers. From the latest running shoes to the oldest Jordans, Harajuku has it all and they're conveniently located in one part of Harajuku.
The entrance to sneaker heaven is at the end of Takeshita street and across Meiji Dori. Murasaki sports will be straight ahead with the small alley to it's right. Immediately to your left will be Atmos and Chapter. Chapter has been in Harajuku for decades and has a large collection Nike, Adidas, Puma, New Balance, and Vans. Chapter usually stocks old colorways for higher prices while Atmos sticks to the newest releases. Atmos is fairly new to the area but it's a chain store with multiple locations. They also stock t-shirts and jackets on the bottom floor. Both Atmos and Chapter will have the latest designs out on release day but be careful as the abundance of tourist in the area will surely guarantee they will be sold out very quickly.
Continue down the street and make the first right, and immediately you'll find more stores. The LA based Supreme store is located on this street as well as the equally famous Undefeated. Both feature newer sneakers but not the variety of the other stores. Supreme and Undefeated are typically more expensive as well. Billy's sits across the street from both and is the newest among the shops. A very chic and spacious store with sneaker styles that rival Atmos and Chapter.
But the holy grail of the sneaker shops is Kicks Lab. Kicks Lab has an enormous collection of Nike, Puma, Adidas, and virtually every sneaker brand to out there. And the collection that sets them apart is their Jordan collection. A massive wall with almost every Jordan shoe that are guaranteed authentic. But be mindful that these shoes come at a very high price as most of them are dead-stock and are not being produced anymore, some ranging up to 40,000 yen. But even if you don't intend on buying shoes that expensive it is still worth the trip to visit the store. In an ever growing market than depends on online sales and aftermarket trading, finding an actual store that sells dead-stock sneakers is incredibly rare and equally beautiful. They do sell tax free to international shoppers - however little it may help.