Dashboard Contact Form

Yes. You can do this by adding a dashboard widget with a contact form function that uses AJAX to submit the form.

function myplugin_dashboard_widget() {
    ajax_contact_form();
}

function myplugin_add_dashboard_widgets() {
    wp_add_dashboard_widget('myplugin_contact_widget', 'My Plugin Contact Form', 'myplugin_dashboard_widget');
}

add_action('wp_dashboard_setup', 'myplugin_add_dashboard_widgets' );