I serve as the President of the Veterans Affairs Commission, City and County of San Francisco.
It is an advisory Commission to the Mayor of San Francisco.
"Possession" is not an issue.
No one 'owns' the Commission.

Some local self-appointed experts insist that there should be an apostrophe...after the first word.
As in: Veterans' Affairs Commission.

I say; incorrect & inappropriate.
No apostrophe needed/required.

What does an actual grammar expert say?

Please send reply to:
noetzelder@gmail.com

I'd agree that possession is not really an issue when it comes to the Commission; the veterans' affairs aren't 'possessed' by the Commission.

But notice that I did use an apostrophe in the last sentence. The affairs 'belong' to the veterans; they are the 'affairs of the veterans'. Therefore, there should be an apostrophe to indicate that possession: Veterans' Affairs Commission.