Is it permissible to sell shirts designed through the Teespring website, which are only printed after purchase, knowing that the website is responsible for printing and sending the product to the buyer, and does the Hadith: "Do not sell what you do not possess" apply to this transaction?

There is no harm in selling the shirt described on the website if its specifications are accurate and remove ambiguity. This is considered a salam (forward) sale and does not fall under the prohibition of selling what one does not own, as that prohibition is specific to selling tangible goods. There is also no harm to the questioner, as he is an agent for the website.