What is the ruling on affiliate e-commerce, which involves a marketer promoting goods they do not own, after which the original website ships the goods to the customer, collects the full price, and sends the agreed-upon profit margin to the marketer?

There is no legal impediment to this work according to Islamic law. In essence, it is an agency agreement with a commission (the agreed-upon profit margin). It is permissible to appoint an agent for a commission because it involves an action for another person that is not obligatory for the agent, thus making it permissible to take compensation for it. If the principal says in the agency agreement, "Sell this for ten, and whatever it exceeds is yours," then it is valid, and the agent is entitled to the excess. This work can also be characterized as brokerage, which is a ji'alah (contract of reward) agreement.