It starts to bring up questions of personal liberty... can I hire someone (TaskRabbit, DoorDash) to do something on my behalf (buy In-N-Out), and act as my agent? And does In-N-Out have any basis for preventing me from doing that?
I would think the answer as of currently, is no. That is less because forcing that restriction would be legal or illegal, and moreso because when you buy food at in-n-out you are not waving any rights or signing any document. If for instance, before every sale of a burger they made you tap a screen saying "I agree this purchase is not for delivery" then yes, the delivery person/service could be sued for tortuous interference. Unless that happens, there is broad restrictions on what companys can do to restrict secondary sales.
The only caveat could be that restaurants are regulated for quality and health. If a lawsuit could prove that the delivery services are actually re-selling the food instead of merely delivering them, there might be a case there. Especially because I highly doubt these services have the relevant licenses to prepare and sell food. Its possible though.
Assuming it's only because they are doordash couriers, and not for any protected reason (race, gender, etc. In CA, even unconventional dress is protected).
Note, however, that if companies conspired together to refuse service to doordash couriers, that would be an antitrust violation known as "concerted refusal to deal".
At the same time, good luck trying to determine which orders are DoorDash...