Unable to load DLL 'Microsoft.WITDataStore32.dll' (TeamFoundation.WorkItemTracking)

You don't need to install the TFS object model/Team Explorer anymore. You can use the NuGet package

Also with the NuGet package, you may run into the same issue that your application cannot find the native DLL.

The text below is copied from https://connect.microsoft.com/VisualStudio/feedback/details/1695433/team-foundation-server-2015-sdk-missing-microsoft-witdatastore64-dll

Microsoft.WITDataStore*.dll is part of the ExtendedClient package, they are native dlls and cannot be referenced in managed project. You will need to manually copy the dll into your bin folder for runtime resolution.

Microsoft.WITDataStore32.dll is in ..\Microsoft.TeamFoundationServer.ExtendedClient.14.83.1\lib\native\x86 Microsoft.WITDataStore64.dll is in ..\Microsoft.TeamFoundationServer.ExtendedClient.14.83.1\lib\native\amd64

(note these paths point to the NuGet package folder)

In the end, the following stackoverflow threads gave me the answer:

where is the tfs 2010 api dll microsoft teamfoundation framework server dll

what do i need to install to get microsoft teamfoundation workitemtracking client

The VS2015 dll's are not compatible with TFS 2013, so don't go looking for the dll's in Microsoft Visual Studio 14.0\ Common7 \ IDE !

You HAVE to install Team Explorer 2013 if you work against TFS 2013

It's small (just 130 MB), and it contains the correct DLLs.

After installation, you need to Browse... for the references here:

C:\Program Files (x86)\Microsoft Visual Studio 12.0\Common7\IDE\ReferenceAssemblies