A quick note to point out that the latest version of the Amazon Kindle app for iOS (6.23) has added a public library filter allowing users to, “view books you’ve checked out from your public library.”
You can not search for library available ebooks using the Kindle app.
For the past eight years Amazon has had a relationship with OverDrive that allows library users borrowing an ebook from a local library (via OverDrive) to wirelessly place the book(s) on their Kindle device.
Other options allowing users to both search for and read OverDrive ebooks via iOS include OverDrive’s app named Libby.