Automator automation to sort fields in Contacts?

The AppleScript scripting dictionary for Contacts has no sort verb, so Automator is out.

Your best bet on sorting Contacts is by selecting those that you want, and drag/drop them onto an open Numbers spreadsheet's first selected cell. Before you launch Numbers, visit System Preferences > Security & Privacy panel. Click the Privacy tab, sign-in, and select Contacts. Ensure that Numbers and Contacts are both checked there — so you will have permssions to perform the drag and drop multiple (⌘=key) of your selected Contacts onto the spreadsheet.

The available Contacts card fields will populate in rows and columns, giving you an opportunity to sort where you want.

I forgot to mention that in Contacts Preferences > General panel you can configure it to automatically sort by First or Last name, but no other fields. In the Mac App Store, you can search on Contacts and that will find several apps for this purpose that may offer free demos from their websites. Perhaps the offer alternative sorting to First or Last name.

Other light reading: Contacts User Guide for Mac - Apple Support