If you install Unity via the web browser prompt on Windows operating systems, the Unity web player installs for the current user only. This allows users without admin privileges to run the installer, which in general is a good thing.
However, this means that to install for all users, you need to run the installer using an account with admin privileges.
These are the full instructions to install the unity web player for all users on Windows:
1) Download the Microsoft Installer for UnityWebPlayer5.1.3f.msi (last updated 28th August 2015) distributed by IntoScience which performs a full install of Unity Web Player version 5.1.3f.
2) Run the downloaded installer file by double clicking it.