Hi there @Armin77!
Don here from Shopify. :)
While there are plenty of services available for you to partner with for your store to print items on demand, they would usually work by having you upload designs in advance, from my experience.
You can check out an in-depth guide to getting this set up for your Shopify store here on our help site.
If instead of uploading designs in advance you wish to pull them in via API, you would need to discuss this with the print-on-demand service you intend to work with.
It might be necessary to have a custom solution created by an Expert Developer, though any solution they create would need to be compatible with the third-party printing service you choose to work with so keep that in mind.
All the best!