Hi,
Great question! Food is of course an important part of student life...!
There is a Tesco nearby (about 10 mins walking or 3 mins on the bus) should you want to go into an actual store- but avoid it on move in day because it is chaos!
In terms of deliveries from a supermarket to Harwich Court, you're completely right unfortunately they won't be able to deliver straight to your flat door. Due to a barrier they can only go as far as the taxi rank by the sports centre- so some carrying will be required. Normally they will call you to let you know that they have arrived and often they are doing multiple deliveries to other flats.
Hope this helps