And today OverDrive announced that their ebooks were compatible with Kindle FreeTime. (They weren't already?) Parents can check out ebooks from more than 20,000 schools and libraries in the US and use them with Amazon’s Kindle FreeTime.
eBooks borrowed from libraries and schools in the Kindle format may be added to FreeTime profiles on a Kindle Fire by following these instructions, or through the “Manage Content & Subscription” section in FreeTime by taking the following steps:
1) On the Start screen for FreeTime, tap “Manage Content & Subscription.”
2) Tap “Add titles to [name of profile]’s Library.”
3) Select “Books” from the dropdown menu.
4) Check the box next to the desired title and tap “Done” in the upper right corner. The title will be added.
OverDrive is the only library ebook provider integrated with the Kindle platform, and that integration is still limited to the US. Tablet owning parents outside the US as well as those who don't have a Kindle Fire might want to consider adding the OverDrive Android app to your kid's tablet. That will probably take less effort since it would only have to be approved once, rather than repeated each time an ebook is checked out.
More details as well as an FAQ can be found in Amazon’s Kindle support pages.