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It is possible to make money from home in the print on demand industry without your own website.
Zazzle is good in that it allows you to run Google analytics on your shopfront. However you can not run the Facebook Pixel on Zazzle so you are not able to put together custom audiences for marketing.
If you just want to try print on demand then you can do it without any money with Zazzle but my advice is to use your own website. If you do that then you don’t lose all your time, effort and money marketing if your Zazzle account is canceled.