Well over a third of the population
When we studied Thucydidies a few decades ago (staff college) the plague in Athens during that time period of the Pelloponesian War (400 BC-ish), during a siege, was a source of some mystery in re "what disease was this?" (was it a novel virus, bacteria based infection?) - a mystery that was being slowly solved thanks to some deep research. (I had some articles on that which are either in the attic or got lost in a move)
What struck me at the time that we studied this stage of the 30 year long war was that almost a third of the population of Athens was thought to have expired thanks to that plague - and Athens still survived. Granted, they didn't win the war.
For your purposes, and considering the social dislocation in Europe during various waves of the Black Death / Bubonic Plague, a threshold somewhere between 1/3 and 1/2 of the popluation succumbing looks like a decent tipping point ... but you'd need to include a few seasonal and climatic factors, as well as tech factors.
Diseases don't happen in a vacuum. Some cultures and societies are more reslient to stresses than others.
Don't assume a uniform distribution of deaths: assume spikes
I'd suggest not assuming a uniform distrubution of death across the population: that assumption would actually make it easier for a social group to whether the storm. I'd look particularly at the problems of how urban population concentrations are far more susceptible to large death spikes and spreading of the disease.
This part of my answer is informed by the book Plagues and Peoples
(which I read about nine years ago). It describes the danger of urban concentrations in spreading disease, but also how the benefits of urban concentrations can achieve the critical mass of brain power (in a cultural development sense), and economies of scale, to figure out how to cure / innoculate against a disease once a certain level of tech is achieved.
Part of your answer to your question form a world building perspective is: what level of tech is the society/culture that I am working with?
As an aside: if you do want to get your hands on that book, make sure that you get the edition that includes the treatment of the HIV/AIDS spread. The original edition was published before that was as well understood as the later edition, which has a section devoted to that particular disease.