First, some background.
Facebook posts that share a link with a nice, in-focus, full-column-wide image do better than posts that show without an image, or with a thumbnail. (Note: this is about sharing links on Facebook, not actual uploaded photos.)
Sometimes the website displays its logo or an ugly or irrelevant photo; sometimes it shares a small, square, thumbnail-sized image. In those instances, on desktop only, Facebook pages (but not individual accounts) can upload their own image instead, making the post more share-worthy and ensuring more people will interact with and see the post.
But what if no image at all comes up?
Very rarely it's because the website itself has no graphic image associated with the page. If that's the case, there's nothing you can do about it.
But usually, there is an image; it just doesn't get delivered to Facebook in that moment.
First, close your Facebook page on your browser. Then go to the Facebook debugger at https://developers.facebook.com/tools/debug/.
Enter the URL of the link you're trying to share.
Here, you can scroll down and see the error messages that prevented the image from displaying. If this is your organization's website, make sure whoever manages your website sees them so they can be fixed.
If it's not your website, or after you've copied the messages, go back up and click "Fetch new scrape information."
Then, for reasons no one can really explain, click it again.
Now, open your organization's Facebook page again. Paste in the URL. This time, it should work correctly!
Remember: If the post looks unappealing, it won't perform as well. Take the time to make sure every shared link has a photo that takes up the whole column, not just a thumbnail, logo, etc. It's worth the few extra minutes to get it right!