Believe it or not, the answer will be in your mortgage and everyone in the community's mortgage. Go get a copy and read the PUD rider. It will say there are 4 things you cannot do: 1) abandon the HOA, 2) have the HOA operate without insurance, 3) amend the HOA documents in areas dealing with protection of lenders and 4) go from professional managemnt to self management.
I currently represent an association that is headless and adrift like yours. The problem is that this condition disqualify ALL the houses from being mortgaged. Almost all lenders inquire into the health of the HOA as a condition of funding. Every week it gets stricter. Do not wait until your association is on a "No Lend" list before acting. I think if you hire a professional management company, a lot of your problems will go away.
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The outgoing Board members should continue to serve until a person can be elected or (more likely) appointed to serve in their place. The power to appoint new Board members in these circumstances is probably contained in the Association's Bylaws. In a pinch, a board can act with as few as two members. The Association obviously needs a functional Board that can take action to address common concerns.
Terminating the association would require a vote of most or all of the owners (depending on what kind of association you are and what is in its governing documents). It is usually more practical to focus on achieving a functional board.