In WIndows 7, when UAC is enabled the Webclient application is installed in the following location:
C:\Users\<username>\AppData\Local\WebClientApps

instead of the ApplicationsPath location specified when installing WebClient:
C:\Program files (x86)\Progress Software\WebClientsApps (default).

An additional key is created in the registry in HKCU\Software\PSC\WebClient\ApplicationsPath with the new 'AppData' value.

Users is part of the (local) admin group.
UAC is set to: Default - Notify me only when programs try to make changes to my computer.

In Windows 10, this happens even when UAC is disbaled

Even though the user belongs to the Admin group, it still doesn't have enough permissions to install the WebClient application in the 'Program Files' folder.

In Windows 10, security has been tightened even more.
Now even with UAC level set to 'Never Notify', the application will be installed under C:\Users\<username>\AppData\Local\WebClientApps.
 

For Windows 7:
Either disable UAC or install the application with elevated privileges:
Run Internet Explorer with the option 'Run As Administrator', and then navigate to the URL which points to the .prowcapc file to install the application.
Or run the Windows Explorer with the same option if the .prowcapc file is available locally.
Then it will be installed in the correct location.

Note that this will be needed for an application update as well, or the following error will occur:
Unable to install/run <application name>. You must have full access to the Application key in the Windows Registry. 
HKEY_LOCAL_MACHINE\Software\<VendorName>\ApplicationName>. Contact your System Administrator.


For Windows 10:
Run the .prowcapc file from an elevated program (started with option 'Run As Administrator')
- Form a web browser. This works with Firefox (Mozilla), Internet Explorer and Edge. Not with Chrome (Webkit).

- The File Manager does not have the option 'Run As Administrator' anymore. This was removed in Win 10.
However, you can still run the Command Prompt (cmd.exe) with this option and run the File Explorer from there.

- Alternatively, start your favorite file manager tool with this option (e.g. Total Commander)

After installation:
Enable the option 'Run As Administrator' in the desktop shortcut properties:
- Open the properties.
- Go to the Compatibility tab.
- Check the 'Run this program as an administrator' checkbox.
- Click OK.
- Run the application.

Reference to other documentation:
WEBCLIENT: Full Access Needed to Application Key in Registry