This error indicates that version 22.214.171.124 of the System.Web.Extensions assembly is missing from the Global Assembly Cache (GAC). Usually this will mean the .NET Framework 3.5 is not installed.
You can verify this by looking in the C:\Windows\assembly directory – see if there is a System.Web.Extensions entry in there, and check what version it is. For it to work, the assembly name, version and public key token need to match that specified in the assemblies list in the API's web.config:
<add assembly="System.Web.Extensions, Version=126.96.36.199, Culture=neutral, PublicKeyToken=31BF3856AD364E35"/>
The solution will likely be to download and install/reinstall the .NET Framework 3.5 Service Pack 1, which can be downloaded from:
On Windows Server 2008R2, the feature can be easily installed using the Server Manager, right-click Features, select Add Features, and select to install .NET Framework 3.5.1.