That’s not the only issue with Amazon cover art & Picard, Picard is using a kind of hacky code to get those images not the official API, and the MusicBrainz website is using the official API, so it happens that the website can show cover art from Amazon while Picard is unable to do it (and not only due to https/http issue that was fixed recently).
That said, using Amazon as a source of cover art isn’t that great, since often Amazon have crappy images, or images not even matching the album (worst case) or the release (most can live with that).
So, if you have an album with missing cover art, better upload it to CAA if you have an image matching the release you have (please no crap).
Picard 1.4dev has also a feature to use cover art stored locally.