Is it permissible to deal with a company that prints a design for a shirt that I promote and is sold to the consumer for a commission, knowing that my work is limited to design and promotion only, or does that fall under the prohibition of "do not sell what you do not possess"?

There is no objection to designing and advertising permissible clothing, and the money received for it is a wage. If you are an agent for a company in designing and selling shirts, there is no objection to taking a commission for the agency, and a fee for the design and advertising. This does not fall under a person selling what he does not own, because you are not a seller but an agent.