Converting a WebClient method to async / await

private async void RequestData(string uri, Action<string> action)
{
    var client = new WebClient();
    string data = await client.DownloadStringTaskAsync(uri);
    action(data);
}

See: http://msdn.microsoft.com/en-us/library/hh194294.aspx

How can I change the method above, but maintain the method signature in order to avoid changing much more of my code?

The best answer is "you don't". If you use async, then use it all the way down.

private async Task<string> RequestData(string uri)
{
  using (var client = new HttpClient())
  {
    return await client.GetStringAsync(uri);
  }
}

Following this example, you first create the async task wtih, then get its result using await:

Task<string> downloadStringTask = client.DownloadStringTaskAsync(new Uri(uri));
string result = await downloadStringTask;