Category search within a widget. Must include the PHP ability in widgets.

wp_search_in_category

<!-- Search Form -->
<div style="margin:10px 0 10px 0;">
<form method="get" id="searchform" action="<?php echo esc_url( home_url( '/' ) ); ?>">
    <table cellspacing="5">
        <tr>
            <td>     
                <label for="s" style="margin:10px;">Search in Categories</label>
            </td>
            <td>
                <input type="text" class="field" name="s" style="width: 100px;"/>
            </td>
            <td>
                <?php wp_dropdown_categories('show_count=1&hierarchical=1'); ?>
                <!-- this shows both the count of posts and hierarchy format - change these to 0 if false -->
            </td>
            <td>
                <input type="submit" class="submit" name="submit" id="searchsubmit" value="Search" style="display: block; position: relative;"/>
            </td>
        </tr>
    </table>
</form>
</div>
<!-- Search Form -->