Use asynchronous client-side javascript methods to handle the
textbox's OnTextChanged event. The script can call the web service
when the Enter key is pressed.

I do not like the idea of calling a web service from client-side
scripting. Too much information is publicly exposed about something
that may need to be a private resource. Instead I would call a page
specifically designed for the task and pass the text in the query
string.

protected void Page_Load(object sender, EventArgs e)
{
 string textboxText = Request.QueryString["TextboxText"].Trim();;
 string labelText - getLabelText(textboxText);

  Response.Clear();
  Response.Write(labelText);
  Response.End();
}

Doing this keeps the async call within the scope and security of the
session.