Our library bought 10 kindles, I setup a transparent proxy which the kindles could connect to fine but when it came to HTTPS this is where it failed. The kindles can browse the web using the experimental web browser they have but they cannot connect to the Amazon store, or send the login details to download books.
So at the moment the kindles have to be taken off site to download the books, joy! I wish Amazon had just allowed proxy settings to be configured on the Kindles.