How do I export contact photos from iCloud?
When exporting vCard from iCloud Web app, the image data is not included in the card. Contact profile picture in such cases can be extracted using the included Web Developer Tools in your Web browser. The procedure is almost similar for all modern Web browsers, I'll use Safari as an example here.

Developer Tools are not enabled by default in Safari. To enable them open Safari preferences by going to Menu bar → Safari → Preferences.. or use the keyboard shortcut Command + ,. Now go to Advanced tab and check Show Develop menu in menu bar.



Now open Contacts section in iCloud Web app and select the desired contact. Once you have selected the contact, right click on the profile picture and select Inspect Element from the context menu. This is generally the last item in the menu.



Clicking on the Inspect Element item will open the Web Inspector, a tool generally used for Web design and development.



As you can see, the Web inspector has opened with a node pre-selected (basically an HTML image tag) corresponding to the profile picture selected. Within the tag, you'll be able to see an item similar to:

href="https://p12-contacts.icloud.com:..."

which is a hyperlink. Simply click on the link. It will display the original image assigned to the contact under the Resources tab in Web Inspector.



Voila! you have the contact profile picture. Simply drag the image to Finder to save it.