My company wants its own website, but would like orders to go to a printer and for them to print and post the t shirts out. We wanted to avoid zazzle

My company (a design agency) wants to make a t shirt shop online for all of our designers to create their own t shirt designs. We are going to give all proceeds to chartiy.

Being a design agency, we really want to design the shop ourself, and then when people order a t shirt, the order gets sent through to a printer who prints the order and posts it to the customer.

How would this work? I can see that a lot of websites offer this service, but including a website with them. We wanted to cut this 'middle man' out, as it would mean less money going to charity.

Can anyone advise the next step? Are there any online shops out there that do the same process that I can contact for advice? How do I know who the best printers are to contact? (UK based preferably)

Thanks!