JM,
You are not on the board.
You are not in a position to really do anything but rally the neighbors and institute a recall or gather enough support so they don't win a reelection effort. However, that will not stop anything except, perhaps, remove some liability from the Association.
The Board, as others have said, can do some things but even those are limited.
They can censure, they can informally warn, they can adopt a policy about Directors posting, they can remove the person from Office (but not from the Board), if they own the site - they can ban the individual from the site or shut it down completely. That's it. However, that will likely not stop what you are describing. In fact, it may escalate it.
I was with Larry when he questioned your statement that the individuals actions may be a violation of their
Fiduciary Duty. Since you did not provide an actual example of what was being posted (but did provide some info), one would have to fill in that blank and it appears that Larry and I took the position of warning others of potential financial issues would, in fact, be fulfilling their fiduciary duty by bringing it to the members attention.
Remember, we do not know what you know and what may be common knowledge to you is not common knowledge to us. Therefore, we take positions and fill in the blanks based on our own experiences. The newsletter I created and ran for three years informing the members of what their Board was doing may have been seen by some as "baseless accusations" or that I had a "personal vendetta. Others didn't see it that way. It's all about perspective on how one views anthers actions.
Since you are not on the Board, you may not have all the details.
Have you tried approaching the individual and ask what the basis is for their position (something many on this site do often)?
It could be that they know something you don't or that they are misinterpreting something you can explain to them.