This is a fantastic question! I am pretty sure the USFA would take over management until some solution could be found.
We (the Metropolitan board) intentionally streamlined and scaled back our involvement and activities a few years ago to only the essentials. This leaves the board with very, very little to do or talk about at meetings. If need be, we could probably get by with a single person managing it if necessary. This cut down on work, bickering, and useless political in-fighting.
You do need an Executive Committee but it can be of equals. Someone has to find venues and set dates for qualifiers. You could have a lottery and call the losers Chair and Vice Chair.
You do need someone to submit the required reports and count the money but you can take turns with that.
The predecessors to the USFA consisted of only a few divisions, which then subdivided to form the multitude we have now.
If the division won't establish someone to do the work, the USFA would then probably assign the fencers to one or more neighboring divisions. You lose your separate qualifiers and undoubtedly have to travel further to fence in tournaments.
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The short answer is to contact the USFA for guidance. Every circumstance will be different. Generally speaking, the USFA strongly prefers that Divisions resolve their own issues.
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