I don’t think any person ‘runs’ it; but they have a President General who serves a three year term:
https://en.wikipedia.org/wiki/Ginnie_Sebastian_Storage
But note that the info isn’t easily available to find on their website.
The DAR accepts donations...
https://www.dar.org/support-dar/giving-opportunities/donate-online
But is not transparent with regard to who’s running it.
Nuff said to my mind.