Nintendo’s latest videogame machine, the Nintendo Switch, is winning fans for both its lineup of popular games and its flexibility—it works as both a living-room console and a hand-held device.

But the real challenge for gamers has been actually getting their hands on it. Production isn’t keeping up with demand in Japan, resulting in blockbuster queues and lotteries there. Over weekends in July and early August, tens of thousands of fans lined up at stores for a one-in-10 chance to buy the $300 console at events that have...