CALABOZO, Venezuela—In Spanish, the name of this town in Venezuela’s scorching central plains means dungeon. For Marta Solorzano, the recipient of a transplanted kidney, being here turned into a death sentence.

With Venezuela’s state-run health-care system in ruins and the country’s economy collapsing, the government last year stopped supplying the pills Ms. Solorzano needed to keep her organ functioning, her family and her doctor said.

Without...