Build a T-Shirt Shopify Cart

We are launching a Restaurants for Ukraine fundraiser. We need a shopify cart that enables a user to select a t-shirt design that we will provide, then choose from one of many sponsor logos. We will likely need to upload new logos often and a simple way to do that is desired. We will need to calculate shipping, and hopefully have a way to automate sending the t-shirt design to our printer and allow them to produce it and ship it.