You can send commands to the hosted web control using the IOleCommandTarget. It's supposed to be equivalent to HtmlDocument.ExecCommand method but for some reason, some commands don't work in the later.

The command you're after is IDM_FIND.

Here is some C# code to run it:

private void SearchButton_Click(object sender, EventArgs e)
{
    dynamic ax = webBrowser1.ActiveXInstance;

    // IHtmlDocument2 also implements IOleCommandTarget
    var qi = (IOleCommandTarget)ax.Document;

    // MSHTML commands group
    var CGID_MSHTML = new Guid("de4ba900-59ca-11cf-9592-444553540000");
    const int IDM_FIND = 67;
    qi.Exec(CGID_MSHTML, IDM_FIND, 0, null, null);
}

[Guid("b722bccb-4e68-101b-a2bc-00aa00404770"), InterfaceType(ComInterfaceType.InterfaceIsIUnknown)]
public interface IOleCommandTarget
{
    void NotDefinedHere(); // don't remove this

    [PreserveSig]
    int Exec([MarshalAs(UnmanagedType.LPStruct)] Guid pguidCmdGroup, int nCmdID, int nCmdexecopt, object pvaIn, [In, Out] object pvaOut);
}