You can insert some javascript which gets the useragent and passes it back to the WinForms application.
public partial class Form1 : Form
{
    public Form1()
    {
        InitializeComponent();
    }
    private void Form1_Load(object sender, EventArgs e)
    {
        webBrowser1.Navigated += webBrowser1_Navigated;
        webBrowser1.Navigate("www.google.com");
    }
    private void webBrowser1_Navigated(object sender, WebBrowserNavigatedEventArgs e)
    {
        var scriptingObject = new UserAgentScriptingObject();
        webBrowser1.ObjectForScripting = scriptingObject;
        HtmlElement scriptEl = webBrowser1.Document.CreateElement("script");
        scriptEl.SetAttribute("text", "function myscript() { window.external.SetUserAgent(navigator.userAgent); }");
        webBrowser1.Document.GetElementsByTagName("head")[0].AppendChild(scriptEl);
        webBrowser1.Document.InvokeScript("myscript");
    }
}
[System.Runtime.InteropServices.ComVisibleAttribute(true)]
public class UserAgentScriptingObject
{
    public void SetUserAgent(string userAgent)
    {
        MessageBox.Show(userAgent);
    }
}