Amid worsening shortages, Venezuela recently reached a milestone of dubious distinction: It has joined the ranks of North Korea and Cuba in rationing food for its citizens.

On a recent, muggy morning, Maria Varge stood in line outside a Centro 99 grocery store, ready to scour the shelves for scarce items like cooking oil and milk. But before entering, Ms. Varge had to scan her fingerprint to ensure she wouldn't buy more than her share....