Docker container is like a kleenex, you use it then you drop it. To be "alive", Docker needs something running in foreground (whereas daemons run in background), that's why you are using Supervisord.
So you need to "redirect/add/merge" process output (access and error) to Supervisord output you see when running your container.
As Drew said, everyone is using https://github.com/coderanger/supervisor-stdout to achieve it (to me this should be added to supervisord project!). Something Drew forgot to say, you may need to add
stdout_logfile=/dev/stdout
stdout_logfile_maxbytes=0
To the supervisord program configuration block.
Something very usefull also, imagine your process is logging in a log file instead of stdout, you can ask supervisord to watch it:
[program:php-fpm-log]
command=tail -f /var/log/php5-fpm.log
stdout_events_enabled=true
stderr_events_enabled=true
This will redirect php5-fpm.log content to stdout then to supervisord stdout via supervisord-stdout.