Hank here from Shopify!
Currently the way the print-on-demand (P.O.D) system works would require a credit-card or PayPal account as required by the app developers. The idea being that you only order the items you require, rather than a select quantity, yet this would be in the form of an up front payment for the items.
One way that this may work for you is if you are using PayPal within your store, and PayPal to pay for the P.O.D app, then this may assist you in terms of the cost the customer has paid, being used to order the item. However, this would be dependant on the app you use, and limiting your store to only using PayPal, if it is available in your chosen country of course.
I will share this guide with you which goes over the business model itself, but there is also another way to look at this, and is what some store owners prefer to do. If possible, you could order your own stock in advance, so you would have the inventory, then you can ship out to customers once the orders come in. That option may not be what you want, but it is just another option to consider.
While we do not have the option similar to what you suggested, different apps may allow you more flexibility, and you can always reach out to the individual app developers to query features. I will share a link to some apps for you to consider in our app store.
Let me know how you get on or if you have any other questions for me!
All the best,