Elle here from Shopify!
My colleague, Nick, answered a similar question in this thread here.
Keep in mind that if you find a supplier, you can always have a private app created by a Shopify Expert to connect your orders directly to the supplier through our API. Alternatively, if they're able to receive orders via email then you can set them up as a custom fulfillment center.
Hope this is helpful and if you have any other questions, let me know! Our support is also available 24/7.
Try Printed Mint. They don't have an app on the store but they integrate with Shopify. They have good customer service as well. Shipping is on the high side, but they have what you want.
Hi! I am in the process of setting up my shopify site, but my other site is fully functional, including an online designer. While I am not set up for drop shipping, this is something I can definitely implement. If you are interested please feel free to contact me.
I found this writing about 8 best print-on-demand apps for Shopify
It's all about the best print-on-demand apps for Shopify stores with many useful features. I hope you can find a suitable app for your problem.
Here recommending some of the suppliers suits your requirements.
Glasswithatwist - The web's best source for custom engraved and screen printed glassware.
Printthis - All Glassware, Drinkware and Mugs can be imprinted with your Logo, Design or image
Gooten Print on Demand - Dazzle your guests with these custom printed multi-purpose pint glasses.
PrintedMint - Specialty wholesale print + fulfillment company that specializes in one-off printing and branded drop-ship solutions.
CustomGlassware - Very first to provide exclusive printed glassware and mugs, which is why we are the most easily recognized website in the business.