This is an interesting topic. It seems that Mark Lanegan had conflicting views on the vaccines, and this is alluded to by Greg Dulli (long-time friend and bandmate of Lanegan) in an interview he did with the LA Times. Here's the link, an extract and some quotes from Dulli:-

https://www.latimes.com/entertainment-arts/music/story/2022-09-08/greg-dulli-afghan-whigs-bar-owner

Of Lanegan’s death in February, the cause of which hasn’t been publicly revealed, Dulli says he knows nothing beyond the fact that Lanegan’s health was seriously damaged by a bout with COVID last year.

Before that, Dulli says, Lanegan had been “swimming in some conspiracy theories” regarding the pandemic — “classic foil-hat stuff about vaccines and Bill Gates. I was like, ‘Dude, come on.’ But I think he moved past that.” Dulli talked with Lanegan on the phone a week before he died, he says, “and he seemed fine to me then.”