If you want the shirts "printed on demand" only when there are orders, you would need to go with
DTG or heat transfers.
Try searching google for contract direct to garment or contract
DTG printing and you should find companies that have
DTG printers who can print on demand for you.
If you want screen printing quality, then you would have to buy minimum quantity orders (varies depending on the screen printer). Then you would hold the printed t-shirts at your home/office and when the orders come in, you would ship out the shirts directly to the customers.