You can automate Web API testing using RESTSHARP & NewtonSoft. There are very simple http calls that you need to make and easy to code. I have attached a sample of what we have done on our project.
If you don't have Visual Studio, the the latest version of Test Studio can be used for your automation. You need to take help from your developers in understanding on how to make calls and create POCO.
In my case, i have stored all the data in an excel spread sheet, parse the sheet, use NewtonSoft to serialize and Deserialize the objects (POCO). I also store all the JSONs for completeness and correctness.
Hope this helps, have attached a sample. You need to include the libraries for restsharp & newtonsoft.
You may use Telerik Fiddler which can also be used to automate WebAPI. This is more simple, there is an article on Telerik site