How to set up Web API Routing for a Proxy Controller?

Here's how I got this to work. First, create a controller with a method for each verb you want to support:

public class ProxyController : ApiController
{
    private Uri _baseUri = new Uri("http://otherwebservice.com");

    public async Task<HttpResponseMessage> Get(string url)
    {
    }

    public async Task<HttpResponseMessage> Post(string url)
    {
    }

    public async Task<HttpResponseMessage> Put(string url)
    {
    }

    public async Task<HttpResponseMessage> Delete(string url)
    {
    }
}

The methods are async because they're going to use an HttpClient. Map your route like this:

config.Routes.MapHttpRoute(
  name: "Proxy",
  routeTemplate: "api/Proxy/{*url}",
  defaults: new { controller = "Proxy" });

Now back to the Get method in the controller. Create an HttpClient object, create a new HttpRequestMessage object with the appropriate Url, copy everything (or almost everything) from the original request message, then call SendAsync():

public async Task<HttpResponseMessage> Get(string url)
{
    using (var httpClient = new HttpClient())         
    {
        string absoluteUrl = _baseUri.ToString() + "/" + url + Request.RequestUri.Query;
        var proxyRequest = new HttpRequestMessage(Request.Method, absoluteUrl);
        foreach (var header in Request.Headers)
        {
            proxyRequest.Headers.Add(header.Key, header.Value);
        }

        return await httpClient.SendAsync(proxyRequest, HttpCompletionOption.ResponseContentRead);
    }
}

The URL combining could be more sophisticated, but that's the basic idea. For the Post and Put methods, you'll also need to copy the request body

Also please note a HttpCompletionOption.ResponseContentRead parameter passed in SendAsync call, because without it, ASP.NET will spend an exremeley long time reading the content if the content is large (in my case, it changed a 500KB 100ms request into a 60s request).