Rumor: Many people think you need a credit card in order to have an Amazon Kindle account.
If that were true, it would make it tough to offer children equitable access to OverDrive eBooks. But it's not. I went to Yahoo and created an email address on the spot, then went to the Amazon "Manage Your Kindle" page and created an account. Using a library card for a dummy patron (who, of course, has no credit cards), I went to OverDrive, logged in, found a non-Penguin book, checked it out, and clicked on Get For Kindle. That took me to Amazon where I logged in with my bogus Yahoo ID and assigned my OverDrive book to my Kindle Cloud Reader. I then opened the book and read the first page. Bottom Line: You don't need a credit card to read OverDrive books on a Kindle.
NOTE: There are ways to do this that will fail. Penguin does not allow wireless transfer of titles they publish to Kindles. You have to use a USB connection, which pretty much means you need a physical Kindle. And if you have a physical Kindle, you probably have a credit card account. Also, the Kindle Cloud Reader is a browser plug-in. It works if you use Chrome or Firefox, but it won't work with Internet Explorer. But neither of these issues contradict the bottom line: you do not have to have a credit card to read OverDrive eBooks on Kindles.