I know this topic is 7 months old, but I’m having the same problem. Except the alt text for my images IS THERE in the media editor on WordPress, but not showing up in the SEO checker – Screaming Frog to be exact.
I’ve only just starting implementing alt text properly for every image, as SEO wasn’t taking as big a focus for us as it is now. But since I’ve gone back and updated every image, it’s not reflected in the SEO checker. I can see in the source code that the alt text isn’t there, yet it is in the media editor. I exported the alt text file from Screaming Frog, and I can see that some images are showing the alt text, even though 90% of the images are now optimized. Usually when I make changes on the site, they are updated instantly on Screaming Frog, but this doesn’t seem to be making a difference whether I add them in or not.