How is it that Steffon Bratheon failed to find a single possible wife for Rhaegar in Volantis “of noble birth from an old Valyrian bloodline”? Surely there was at least one woman who fit that definition? And why go to Volantis when the Rogares in Lys (or another Lysene family) are much closer with more precedent (Larra Rogare and Viserys II)?

We don’t know why Steffon failed. 

There’s a couple possibilities: one is that, given the repeated failures of such unions (the downfall of the Rogares during the Lyseni Spring, Kiera of Tyrosh’s misfortunes, Rohane of Tyrosh involving the city with the Blackfyre Rebellions), the nobility of the Free Cities decided to take a pass on marrying into the Mad King’s family. Alternatively, it could be that Aerys’ specifications were so ridiculously exacting (I’m thinking a mix between really racist and princess and the pea, or some HUGE dowry) that no one could qualify. 

As to why Volantis, “the king sent Lord Steffon across the narrow sea on a mission to Old Volantis.” Not to the Free Cities – specifically to Volantis. Who knows why; my guess is that Aerys liked their apartheid policies.