It can be done and it can be done profitably, it's going to be more about finding people who don't want to spend their time going to the grocery store.
Grocery shopping in a browser UI can be a decent experience but will probably never be as clear and convenient as seeing the actual item, especially for non-prepackaged (meat, fish, produce, floral, etc.)
Convenience - online shopping requires planning ahead and being home in a time window to receive delivery. For the homebound it is no issue, but for most working people it's easier to just stop by the store on the way home.
I already leave my apartment for work at around 8:30 am and, after stopping at the gym on my way home, I'm usually home around 8 pm.
If I make a grocery store trip, this adds probably 45 minutes to an hour, meaning I get home at 9 pm, which sucks.
I've started getting produce through a local CSA-type delivery service, and it's so convenient and easy. I think Amazon Fresh will work well for people who are busy and have the disposable income to spend an little bit more on delivery.
If they could integrate with some popular recipe sites to add little "Buy this on ..." buttons to each item in the ingredients list it'd be even better.