How did Swain get Vastayan blood?

Distant ancestry, blood infusion, made into one....could be any of these things.

Based on the Rakan interaction he was "given" it or retrieved it somehow. I highly doubt he was born with it but only time will tell.

Well, just from the way Rakan had said the line I assume that Swain was once on more amiable terms with the Vastaya in comparison to now-- I'm getting more shamed/shunned vibes from the entire scenario. Perhaps it could be linked back to his relationship with Beatrice and how their bond works? In addition to that, remember that Rakan mentions Swain choosing to be blind; something to do with returning to rule Noxus perhaps and perhaps even the Ionian Invasion? Considering he did come to rise to power in an incredibly short amount of time. I can only speculate so much unfortunately ;; only the Noxus update can clarify things.