https://www.vbulletin.org/forum/external/2015/08/1.jpg

This was requested, I shall share it with the rest of you all as well.

Find:

<tr>
<td class="vbmenu_option"><a href="search.php$session[sessionurl_q]" accesskey="4" rel="nofollow">$vbphrase[advanced_search]</a></td>
</tr>

Add below:

<tr>
<td class="thead">Search Site / Google</td>
</tr>
<tr>
<td class="vbmenu_option" title="nohilite">
<form action="http://www.google.com/search" method="get" name="search">
<input type="hidden" name="forid" value="1"/>
<input type="hidden" name="ie" value="ISO-8859-1"/>
<input type="hidden" name="oe" value="ISO-8859-1"/>
<input type="hidden" name="cof" value="http://www.yoursite.com/forum/images/autosportnc-yellow/logo.gif;S:http://www.yoursite.com;FORID:1"/>
<input name="domains" type="hidden" value="yoursite.com"/>
<input type="hidden" name="hl" value="en"/>
<input type="text" class="bginput" name="q" size="25" maxlength="255"/>
<input type="submit" class="button" value="Go" /><br />
<input name="sitesearch" type="radio" value="http://www.yoursite.com/forum" checked="checked"/> YourSite.com<br />
<input name="sitesearch" type="radio" value=""/><a href="http://www.google.com/search"><img src="http://www.google.com/logos/Logo_25wht.gif" border="0" alt="Google" align="middle"/></a>
</form>
</td>
</tr>

</table>
</div>


Change all instances of yoursite.com to your forum's URL & Enjoy!