Automatic feeders are worthless in my opinion; they frequently fail and are difficult to regulate.

An adult or larger juvenile will be fine without food for seven days; even a small juvenile would probably be fine if you made sure to feed him extra in the days before you leave. You could add a bunch of live aquatic prey (guppies, rosy reds, ghost shrimp) or live plants (anacharis, cabomba) and let your turtle snack on those while you are gone.