Can you add a profile picture to a IMDb page posthumous?

Well, doing so requires claiming the corresponding IMDb name page and maintaining the claim, in the sense of being able to manage (on a whim) certain aspects of the page. A person who would've been able to claim it while the page's subject was alive would probably still be able to do so afterward (unless maybe there is some kind of a probate-style dispute). In order for you to be eligible to claim a page about somebody other than yourself, you'd have to be a parent, guardian or agent of that person or be a representative of the person's estate.