alt Hacker NewsIt's probably rather difficult and risky to enact vigilante justice against someone who's in prison.
I think the problem is with places where they don't have life sentences at all, but rather let murderers back out into society after some time. I don't know if vigilante justice is a problem there in reality, but at least I can see it as a possibility: someone might still be angry that you murdered their relative after 20 years and come kill you when you're released.
The reference to vigilante justice may be about killing a suspect before they're imprisoned or even tried, such as when a mob storms the local jail. The theory is, if people believe only death can bring justice, and the state doesn't have the death penalty, then the vigilantes will take matters into their own hands. Ergo, the state should have the death penalty.
Having recently done an in-depth review of arguments for and against the death penalty,[1] I can say that this argument is not prominent in the discourse.
[1]: https://fairmind.org/guides/death-penalty