My name is Kassey and I'm a Guru here at Shopify. If you are to take a look at the Shopify Apps page, there are quite a few apps that would allow you to design your products, have them printed by a company and fulfilled by the company as well. I went ahead and grabbed you two links for apps that I think would work nicely for your shop. Be sure to check out: Printful and Printify!
Have a great day!