Normally when a person makes a gift to a nonprofit, the gift become the property of the nonprofit and the donor loses control over it.  The donor may be able to suggest how the gift is used but has no legal right to dictate the response.  The fact that you apparently saw the frame and the plaque without complaining before you left the organization suggests that you made no claim that the flag was yours and not the organization’s.

If you are not able to just call someone, you might write a letter asking to get the flag back because it is so meaningful to you.  The organization may want to de-escalate the situation and agree to its return.  If that doesn’t work and you know a lawyer who would be willing to write a letter for you pro bono, a second appeal on the same basis might possibly help.  But if you don’t have some facts on which to argue that you never gave the flag to the organization and have a right to get it back, I don’t think I would throw good money after bad by buying a lawyer letter.  A lawyer is not supposed to make a demand without facts to back it up and I don’t see anything in your question that would justify a demand for the flag.

Use this as a learning opportunity if you are ever willing to volunteer again.