In Seattle, Amazon is testing Amazon Fresh for groceries. Amazon Fresh has their own delivery infrastructure with their own trucks and drivers.

They also deliver other things using this system. I received a CD via Amazon Fresh on Sunday, that I'd ordered on Saturday, whilst still paying for Tuesday delivery (free with Prime)

If Amazon can make this scale, they are absolutely competing with them directly. If Amazon can't make this scale, then it's unlikely to be a profitable business model for anyone else.