I think you stand a better chance of being able to walk-up at Captain's Grille than you do at an in-park venue or one of the more popular resort restaurants. Still it might depend a little on when you're going in June (weekend or weekday, is there any major event happening?) and at what time you want to eat (prime dinner hour or early/late). As long as you're flexible, I'd guess you'd be okay, but if you have to eat a specific time, I would make an ADR.