When Kenyan farmer Waki Munyalo’s crop of lentils shrank by three quarters last year because of unusually low rainfall, disaster was averted thanks to a simple remedy that was previously unavailable to most African farmers — insurance.

Munyalo, a 38-year-old mother of three, had taken out an insurance policy with Kenyan start-up Pula, which aims to reach a vast, untapped pool of African farmers grappling with new disruptions linked to climate change...