Your Printify API key doesn't go anywhere because there is no ready-made integration available within Odoo.

Unless Printify on the other hand has an actual Odoo integration, and if there is none provided in https://apps.odoo.com or elsewhere by the community, you will most likely have to invest into a custom integration on your own.

Alternatively, starting from Odoo 18, there is an integration for Gelato available (https://www.gelato.com, https://github.com/odoo/odoo/tree/18.0/addons/sale_gelato) which seems to be suitable for similar use cases (custom print jobs on demand) at least.