The resource cannot be found (404) means that you are not using the correct URL to access it, your resource is located at some other location. Either you configured the application incorrectly, or you need to check the URL being accessed. What they are, only you know.
One thing that I can surely tell you is that you need to use
HTTP POST request, as your marked the function as POST acceptor, for that you need to write this in the form; or using JavaScript change the request method to POST, which will be tough. Try this,
@using (Html.BeginForm("SendSMS", "Home", FormMethod.Post))
This will send the request to the server upon submission using POST verb;
ASP.NET will accept it this time. Note that it will only send the request, but no data will be read on the server-side, this makes me think a lot of things, such as why do you even want to do this? Why not create a simple interface or a backend class that handles these, and you only do,
smsHelper.SendSms();
That will be much safer way of doing this. It will also help you modify your code later one, if you want to make any further changes.