It is difficult to estimate the number of people employed in food delivery. There are full time drivers who work for, for example, pizza restaurants; there are people who work for restaurants in other capacities that sometimes go on delivery runs and there are people who work on a very part time basis through Uber-like food delivery apps. As precarious as those jobs are, they seem safe for now.
The food delivery robot shown in this video from Wired is more of a novelty than anything. The robots will have to get a lot better and then the price will have to come down before they are practical for most restaurants in most locations. Still, with autonomous (self-driving) vehicles coming on quickly it’s entirely possible that businesses will use self-driving cars for the job and simply ask customers to come out to the car. It’s also likely that an autonomous vehicle designed for city streets and specifically for food delivery or other courier work will be developed at some point.
Delivery and courier jobs generally appear to be in a position to outlast taxi (uber) and cargo driving jobs. In one case the cars are ferrying passengers who complete the transaction when they exit the car, in the other there are people waiting to unload the vehicle at its destination. Getting the pizza to the house will be a relatively simple task, getting it to the door (especially if there are stairs involved) is a different task and consumers will generally resist taking on extra work themselves unless there is a discount involved.