I think Scott is currently experimenting with building/running a heavily stripped-down subset of FlightGear to specifically target OpenGL Core profile, which could be considered one of the prerequisites to "eventually" target Vulkan, but as has been said, it's unlikely to be relevant/feature-complete for several years to come (I would not even be surprised if that means 3-5 years):
http://wiki.flightgear.org/OpenGL#Statushttp://wiki.flightgear.org/Vulkan_Scene_Graphhttp://wiki.flightgear.org/FlightGear_and_OpenGL_ESObviously, "VSG" is not event yet a thing - so there's that, too.
Other than that, there are quite a few legacy features that would need to be ported/updated.
For instance, the built-in GUI is in the process of being replaced by a Qt5/QQ2 runtime:
http://wiki.flightgear.org/PUIThe time frame here really depends on how many people are interested/involved in making this happen obviously.
However, it is worth noting that the original FlightGear SSG/OSG migration was initiated back in 2006 and has never been fully completed, i.e. we are not even yet using OSG "fully". Thus, migrating to yet another scene graph implementation seems a little far fetched, especially given all the other "projects" that people pursue.
Also, the initiator of the OSG migration actually laid out a roadmap to encapsulate scene graph specific APIs to help pave the way for future migrations if the need arises, but his advice was basically ignored by other core developers at the time, so Vulkan support is certainly going to take some time and work.