I did a search for keywords:
convert coordinates to us zip code
A fair number of websites came up in the results, including this one on GitHub:
API for converting coordinates (longitude, latitude) to United States zip codes. - johnedmonds/coordinate-to-zip-code
github.com
There are probably others that provide the same service, maybe even offering a web service.
I'm pretty sure the lat/lon coordinates of a photo are readable using the Spotlight metadata interface. Metadata is readable with the Terminal command 'mdls'. You could do a few tests on photos to find out.
Once lat/lon has been extracted for a given file, it can be converted to a ZIP code, and that can then be attached to the file, either in conventional metadata, such as the com.apple.metadata:_kMDItemUserTags xattr, or in a custom xattr. The 'xattr' command can read or write xattrs.
The com.apple.metadata:_kMDItemUserTags is visible in Finder's Get Info pane, in the textbox at the top, where comments can be entered.
I'm not sure how to get Spotlight to read a custom xattr, but maybe there's an existing one related to street address that can be repurposed, since ZIP codes are primarily designed for mail delivery. Or maybe there's some photo-related setting or attribute that can be used to hold a 5-digit number.
If the data is present in the Spotlight metadata, then it should be searchable with Finder searches. It may also be searchable with other apps, such as Photos.
This seems like the basis for a useful 3rd-party Mac app, or even an iPhone app, so it might be worth searching the relevant app store.