Hello Roxy,
I was having the same issue on a site I was working on. The pet finder api as is. I reached out to their support team and they provided me with the following info:
The api provides images in http format. If you must have them in https then you must modify the format as described.
Example URL provided from our API:
https://photos.petfinder.com/photos/pets/42993943/1/?bust=1539299405&width=300&-pn.jpg
Change it to this format:
https://dl5zpyw5k3jeb.cloudfront.net/photos/pets/42993943/1/?bust=1539299405&width=300&-pn.jpg
I was able to make edits to the plugin in order to get the photos to show on a site that uses ssl. I will am trying to see if I can do a pr request for the plugin for the change. Please let me know if you would like assistance making this change.