Hey!
I can answer this pretty well as I grew up and lived in a pretty car dependant Minneapolis until I was 23, where then I moved to NYC with no car in 2018 and have lived here ever since.
The TL;DR to this question is that you transport everything in a grocery bag on person, but the longer answer is that your buying and cooking behavior changes.
Back in Minneapolis I relied on buying in bulk, since I wanted to limit the number of trips in the 15-20 min drive between my apt and the Costco. Variability with the weather affected this too, as I would buy extra if it was in the winter time. I'd make this trip by car around 2-3 times a month. This also affected my buying and cooking decision making as well. Buying groceries first then figuring out what I wanted to cook.
Once I moved to NYC, I would always have a grocery store several blocks away from me. At most being a short 5-10 min walk. This changed my habits as I always had a grocery store I could quickly pop in without having to think about traffic, my car, etc. So although I would go more frequently (~ 2 times a week), I would also find this a lot easier and would buy less.
Nowadays, whenever I think of wanting to cook something, I either head over to the store on my way back to the office, after the gym, etc. and then cook that very same day what I bought.
In my mind, that big fridge I used to restock with my Costco runs has been replaced by having that quick grocery store within walking distance.
Purely anecdotal, since I know some other people in the US may be living different than a single guy living in an apt in NYC, but this is also how it is in many cities I've traveled to in Asia and Europe.