I assume your goal is to have the “sold out” product show up on the store but not allow it to be added to the cart.
Without any modifications to the theme you could enable inventory management for the respective products, set their stock to 0 and disable overselling. That way they would show up as sold out and could not be added to the cart.
If you can’t enable inventory management for these products for some reason, the theme would have to be modified so that the respective products could be tagged as “sold out” and would show up as if they had an inventory of 0 even with disabled inventory management.
The PoD app is probably setting all the product properties and disabling inventory. You could still try to enable inventory management for a test product or ask the support of the app providers if there is a way to achieve this.
If you can’t work with inventory management the theme would have to be modified. It’s not a huge change, but the exact steps depend on the theme you’re using. I would recommend hiring an Expert.