Amazon is investigating Uber-like delivery options [1]. If they decide that there's anything there (and I'm not totally convinced that there is), they'll build the service themselves and not pay Uber for the privilege. And if the service works well for themselves, you can bet they'll open it up to other people to use for a price, as they have for all of their internal infrastructure projects.

[1] Source for this claim: I worked at Flywheel, an Uber competitor. Amazon had a program to have our drivers deliver packages, and got fairly close to acquiring us, before deciding they would rather not. This was a few years ago at this point, information is not current, and I have no idea about the present state of Flywheel, as I do not work there any more.