For Asher Anidjar, the arrival of fall isn't marked by turning leaves or a chilly breeze, but a steaming seasonal drink. Recently, though, when he headed to his local Starbucks for a Pumpkin Spice Latte, he left with a bitter taste in his mouth. They were out of the special sauce that gives the treat its distinctive autumnal flavor.

"I just left, depressed," said Mr. Anidjar, a 26-year-old commercial real-estate analyst who lives in Manhattan.

The...