How to send request from XMLHTTP without waiting for the response

You might consider writing a bit of .NET wrapper code to consume the web service. Then expose the .NET code as a COM object that the ASP can call directly. As you've seen, there is no tooling to help you in classic ASP, so consider using as much .NET as possible, for the tooling. Then, use COM to interoperate between the two.

You can use this component for http-requests like "POST", "GET", "DELETE" etc.

To create the object:

<%
    Set objXML = Server.CreateObject("MSXML2.ServerXMLHTTP")
%>

To send data using method "GET":

<%
    objXML.Open "GET", "http://mvp.sos.state.ga.us/?some=querystring", false 
    objXML.Send ""
    Response.Write objXML.responseText
%>

Note that Open method has 3 parameters: HTTP method, URL, asynchronous call.

Note that Send method on a "GET" ignores its parameter. (In this case we are passing parameters via the URL.)

To send data using method "POST":

<%
    objXML.Open "POST", "http://mvp.sos.state.ga.us/", false 
    objXML.Send "username=htbasaran&password=somepassword"
    Response.Write objXML.responseText
%>

Note for "POST" that Send method passes parameters in key-value pairs format like: key1=value1&key2=value2&so=on... or any other data like XML, JSON, etc.)

These are the basics of this component. If you need more information, you can check microsoft's docs page out.

An example code for getting form values and sending them using xmlhttp post method.

<%
    ' getting form values
    my_uname = Request.Form("username")
    my_pword = Request.Form("password")

    ' creating object
    Set objXML = Server.CreateObject("MSXML2.ServerXMLHTTP")

    ' sending variables to an external site
    objXML.Open "POST", "http://www.sitename.com/login.asp", false
    objXML.Send "username=" & my_uname & "&password=" & my_pword

    ' Assuming that successful login will return response "Ok"
    ' writing the result to the client.
    if objXML.responseText="Ok" then
        Response.Write "Login Successful!"
    else
        Response.Write "Login Failed!"
    end if
%>