The problem is in your tpl_modules_product_listing.php template file.
I can duplicate the problem on your site by clicking the Animals->Horses category, and then entering anything in the search sidebox and clicking Search.
The problem is repeatable whenever you have the product-listing page open.
Thus, the tpl_modules_product_Listing.php file must have the closing </form> tag missing from it.
In v1.2.7, the last 7 lines of the file are:
PHP Code:
<?php
} // show_bottom_submit_button
?>
<?php if ($show_top_submit_button == 'true' or $show_bottom_submit_button == 'true') { ?>
</form>
<?php } ?>
</table>
I suspect maybe you have some sort of addon installed or may be some switch setting in the admin area that's causing this.
At any rate, the problem doesn't exist in v1.3.x