Seems like in any market where Amazon can reasonably compete with a brick & mortar, Amazon will win... Consider how expensive real estate is these days, especially in urban areas (and how much "millennials" like delivery). It's a solid formula.
It's not just millenials in this case. I am an GenX-er with a wife and two kids. We are planning to switch to a mail order pharmacy (Pill Pack or Costco -- dependent on cost) in the near future. My wife, one of our daughters, and our dog (on a human med) taking regular meds. Even being a 5 minute drive from a brick and mortar pharmacy, and it is still a PIA. None of the scripts are on the same schedule so we make multiple trips. Inevitably, you get in line behind someone with a refill or insurance issue turning a quick pick into a 20 minute wait. A service that syncs up script refill schedules and works with doctors and insurance companies on refills renewals is a massive time saver and stress reducer.
But think about how much it costs to have the item physically delivered. There is no way that Amazon can make money on a $10 item that they need to deliver in a two day time frame.