Is there a way to download the Steam app's Update News art?

Since this method is not the intended user level access of Steam resources, please consider these steps as fitting only for discussion. Since Steam has made the interfaces and the window especially to not support making copies from there, I'm also not suggesting to do it.

However, yes, you are correct that all these are stored on your PC. For the case of a Windows10 OS, you will find all the "Update News" pictures at the local temporary folder:

c:Users [YOUR LOGGED IN USERNAME] AppDataLocalSteamhtmlcacheCache

However what you find here is a list of objects without extensions, which do include the pictures, but also other elements too, like elements from Friends list. You may want to somehow filter out the irrelevant ones.

So as best effort, you can do the following on same day when it is available on News:

0. Close steam if it is running (not running on tray icon, but really shut down the app) (this step might not be necessary, but this is the way how you can do it for sure)
1. From the listed items on the path above you may delete all "f_00..." files (I suggest to not delete the "index" and "data..." files)
2. Start steam and log in. Depending on your settings, Store might automatically open up, or not, for this case the better if sooner opened up, because: these freshly downloaded files will have their timestamp (as of now like 12:57 here).
3. You wait for the next minute (or so), and hit Update news button. Depending on the browsing history, you may get either the normal pictures per page, or a broken reference icon, which you can easily remediate with a Next> then <Previous button, so it will (down)load it.
4. Now check back to the path, you will have some "f_00..." files (1...20...or so) new timestamp (like 13:02) there. These are the relevant files now.
5. Up to your choice, you may analyze or rename them with a tool or manually doing it, renaming them to .jpg extension and checking for the one you wanted, you will eventually find it. Alternatively (and bit improving) that Malady suggested in the comments, you can just drag&drop the files 1-by-1 into a browser window, that will also show you the file content.

You have some chance to find it still on later days if this cache folder was not updated with following days Update News.