Well, it depends, really, since Conan's possessed of a roughly pre-Classical moral code-- like with a lot of mythical Greek heroes there's a whole mess of shady behavior that is totally fair game to any self-respecting wanderer. So I have a really hard time seeing Conan being vengeful to the point where it is self-destructive. As you pointed out, what few rivals he had a history with in Howard's works were typically just other mercenaries or whatever and he never went out of his way to kill someone just because they were on the opposing team. Hell, in some cases like in People of the Black Circle he was pretty willing to just go ahead and work with whoever could get him ahead even if they had tried killing him earlier-- Kerim Shah basically said "We can sort this out (read: kill each other) later" and Conan was down with that plan.
On the other hand, there were some old-school taboos that could get your shit ruined by a pissed-off Cimmerian if you weren't careful. If there's one thing that cuts across ancient cultures, it's the notion of hospitality. Offering someone shelter and then selling them out just isn't done, and that's why Conan happily went out of his way to kill an innkeeper that had earlier tried selling him to cannibals in the story Shadows in Zamboula. And then there's the case of Balthus and Slasher, a soldier and a dog featured in Beyond the Black River. They impressed him enough that upon hearing of their deaths Conan announced he would take the heads of 10 Picts for Balthus and 7 for the dog, who he considered a better warrior than most men. So I could see a vengeance plot working for Conan, but it'd be sort of a hard thing to make very interesting given that the few times he takes vengeance it's not even really a particularly emotional thing for him, it's just kinda how Cimmerians roll.