How to add a shortcode to an HTML image tag

Use a separate function to enable shortcodes in the text widget because trying to do so within the shortcode function itself is too late.

// Enable the use of shortcodes within the text widget.
function wpse249090_widget_text_enable_shortcodes() {
    add_filter( 'widget_text', 'do_shortcode' );
}
add_action( 'init', 'wpse249090_widget_text_enable_shortcodes' );

// Do TimeStamper Shortcode ( timestamper() ) 
function timestamper( $atts ) {
  $time = current_time( 'timestamp' );
  return $time;
}
add_shortcode( 'timestamp', 'timestamper' );