I think this is the block from the navbar template that does the search box:
HTML Code:
<vb:if condition="$vboptions['enablesearches']">
<div id="globalsearch" class="globalsearch">
<form action="search.php?{vb:raw session.sessionurl}do=process" method="post" id="navbar_search" class="navbar_search">
<vb:comment><input type="hidden" name="s" value="{vb:raw session.sessionurl}" /></vb:comment>
<input type="hidden" name="securitytoken" value="{vb:raw bbuserinfo.securitytoken}" />
<input type="hidden" name="do" value="process" />
<span class="textboxcontainer"><span><input type="text" value="" name="query" class="textbox" tabindex="99"/></span></span>
<span class="buttoncontainer"><span><input type="image" class="searchbutton" src="{vb:stylevar imgdir_button}/search<vb:if condition="$stylevar['textdirection'] == 'rtl'">_rtl</vb:if>.<vb:if condition="(is_browser('ie') AND !is_browser('ie', 7) AND !is_browser('ie', 8))">gif<vb:else />png</vb:if>" name="submit" onclick="document.getElementById('navbar_search').submit;" tabindex="100"/></span></span>
</form>
<ul class="navbar_advanced_search">
<li><a href="search.php{vb:raw session.sessionurl_q}" accesskey="4">{vb:rawphrase advanced_search}</a></li>
</ul>
</div>
</vb:if>
So if you remove that from the navbar and put it somewhere else it will probably work. (I tried putting it at the end of the header template and it worked although it didn't look right, but I'm sure that just has to do with where I put it).