From their international FAQ:
“To terminate your account, simply send an e-mail to Customer Support (http://zazzle.custhelp.com/app/ask/~/how-can-i-delete-my-zazzle-account%3F) that contains your username and the email address associated with your account. Once you’ve done so, a member of our Support Team will disable your account and remove any products both public and private. The account will be removed fully when our system processes the next batch of account removals.”
For international Zazzle customers there is no mention at all about this termination service. A shame.
Direct removal link: http://zazzle.custhelp.com/app/ask/~/how-can-i-delete-my-zazzle-account%3F
Helpful Trick: Removing online data
Helpful Trick: Remove third-party logins
If you signed up for Zazzle using a third-party login such as Facebook or Twitter, you probably want to revoke all access from Zazzle. Click here to find out how.