Yes, just extract the form code and put it in your page, and make sure you reference the file search.php correctly in the form action:
<TABLE width="100%" height="1" border="0" cellpadding="0" cellspacing="0" align="right" valign="top">
<FORM method="POST" action="search.php" name="productsearchform">
<INPUT type="hidden" name="simple_search" value="Y">
<INPUT type="hidden" name="mode" value="search">
<INPUT type="hidden" name="posted_data[by_title]" value="Y">
<INPUT type="hidden" name="posted_data[by_shortdescr]" value="Y">
<INPUT type="hidden" name="posted_data[by_fulldescr]" value="Y">
<TR>
<TD align="right">Search<INPUT type="text" name="posted_data[substring]" size="20"> </TD>
<td width="10"> </td>
<TD width="23"><A href="javascript
: document.productsearchform.submit()">Go</A></TD>
</TR>
<TR>
<TD colspan="3" align="right" valign="top">
<A href="/equine/search.php" class="SmallLink">Advanced Search</A>
</TD>
</TR>
</FORM>
</TABLE>