The WebPlayer is only required for the old WebPlayer build. Back then users had to download the plugin to run the game in browser. This is no more required with the webgl build (it just ships a huge build…).
When you make standalone for PC/Mac/Linux, nothing is required as long as the user has not been screwing around too much and removed some vital libraries from his computer.
You do not need anything to build a standalone build. For Android and iOS, there are some additional restrictions and libraries needed. For standalone, it goes as is.