SEOUL—Just ahead of 1 p.m. on Friday, Son Jae-son marched to his neighborhood pharmacy, trying to get a prized item in South Korea: his weekly ration of two facial area masks.

South Korean health and fitness officials have urged all citizens, even nutritious types like Mr. Son, to put on filtered masks, triggering the product to disappear from shelves. With his extended loved ones down to just a handful of masks, Mr. Son’s heart sank when he saw that the drug store—which had posted a sign indicating it would have clean stock at 1 p.m.—had already attracted a sizable line….