For the urbanites among the ancient Romans, food vendors (and even things like grab-and-go street vendors and fast food restaurants) were a normal part of life, since their apartments generally didn't have personal kitchens.
I can't find a source for it, but I have to imagine they had at least some personal food delivery services.
Why would you feel that someone who makes your food is beneath you?
I doubt either applied in Roman times. I'll stand corrected if:
a) There is evidence people submitted food schedules where the food could be delivered on a regular basis, or
b) There were evidence of a pre-telephone method of instantly transmitting an order, or
c) It were a practice to send out a messenger to place and bring back the order for the non-wealthy.
http://www.telegraph.co.uk/history/pompeii/9850077/Pompeii-e...
So I imagine it is possible many would do some kind of deliveries as well.
Pizza only came to be much later, when the tomato got introduced to Europe, from South America. And when people eventually found out that many varieties of tomatoes are not only not poisonous but also well tasting. :)