Kelly du Plessis, CEO of nonprofit organisation (NGO) Rare Diseases, will pay her nine staff members for the last time at the end of this month. And then “we’re done”, she says.

Du Plessis’s organisation, which has helped patients with overwhelming illnesses gain access to expensive medical care for the past seven years, is luckier than some other NGOs. Unlike many, it has at least had a three-month salary buffer, which has now run out...