I am starting a food delivery business in my city. We deliver food to your door from local food stores in under an hour. Should I go door to door to promote my business or should I stick with internet conversion?

Of course better to use both ways, but in delivery food business personal contacts is a better promotion than any other ways. Few announcements in internet needed and posters from door to door will bring to you a good result)

That is for sure, and you can get some real feedback from your core potential customers.