Amazon's retail store card situation isn't quite as hands-off for parents as Apple's. You need a credit card to create a new iTunes account, but afterward you can disconnect it and kids can be self-sufficient. They might take some birthday cash, buy an iTunes card just about anywhere, enter the code, they're off. I only get pulled in to help with things like backups and migration between devices.
The tween who uses my Amazon account for her Kindle has hit multiple little headaches I had to resolve for her. Example: Amazon's Kindle gift cards aren't as easy to find as iTunes ones, but she did find one at a local store. It turns out you cannot just redeem them directly on the Kindle and go shopping. You have to visit your account on Amazon's web site, add the gift card there, and then it's available as a credit for Kindle purchases. It was not as smooth of an experience, and I have yet to figure out how to make her self-sufficient on the device. If you give the kids enough account info to add a gift card, they'll get full write access to all the account info. That's a bad plan.