I'd approach this by having javascript in the browser calling the service on the local machine. You'd probably want to have some preliminary calls to ensure the service is there and ready, with messages to the user if there's a problem.
jQuery makes ajax very easy. You'll want to do some reading jquery.com. Grab the js library, and check out the ajax documentation.
Your code will look something like this (and this is completely untested)...
<script src="../Script/jquery-1.9.0.js" type="text/javascript"></script>
...
<script type="text/javascript">
function readSerialBuffer() {
$.ajax({
url: "http://localhost:12345/readSerialBuffer",
success: function(data) {
$(".serialdata").val(data);
}
});
}
</script>
<asp:Textbox runat="server" id="txtSerialData" class="serialdata" />
<input type="button" onclick="readserialbuffer();" />
Something like that. The idea is that the button click runs the ajax function to request the weight, and when that call succeeds, it loads the value returned into the textbox. I haven't tried running this... it's just off the top of my head.