The closest analog we have here is Instacart. You pay a yearly fee for express delivery, and you get to pick from several local stores to get your stuff. Groceries, Target, you can even get Costco without a membership of your own.
The drawbacks are selection and price. On top of express (prime-like pricing) there's also tipping which adds at least 15%. We do it mainly for groceries and some odds and ends, but for sure it's not even coming close to Amazon-level disruption.
And then of course none of these places have Amazon's computing services, lol. That's a whole other sport.