It requires completion of the fate at the time of purchase. its a conditional sales item. The vendors get new items when the fate is completed until the next time the chain starts.
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Related (in case nobody has ever done these chains), the Breaching/Clearing the Hive FATEs in South Shroud also open up a vendor in that area (and actually clears out all the enemies in that little area so you can explore). This is where you can find the Baby Raptor minion.
Bit of a shame that there's not more of these FATEs that actually have an impact on zones/NPCs.
You can look at your list of known songs under your logs. Songs you dont have are greyed out and tell you where to get them.