I have installed c# support for vscode(version is 1.15.0) and created a HelloWorld project via dotnet new console.
Now in Program.cs
System.Web.Extensions is part of full .net framework . If you want to serialize and deserialize object,You can use Newtonsoft.Json,
#using Newtonsoft.Json
....
JsonConvert.DeserializeObject(json);
Update
Just get package name and version number from NuGet and add to .csproj then save. You will be prompted to run restore that will import new packages.