Are you going to Disneyland (California) or Disneyworld (Florida)? Disneyland is 15 minutes up the road from me.
You'll get away with wearing either trainers or flip-flops. The whole beach / surfer scene here means flip flops are pretty popular. But as others have said, you'll be doing a lot of walking and it can get crowded when the parades or on. Having some fat Texan who had steak for breakfast stand on your toes with flip-flops on doesn't bear thinking about
If you are coming to California then it'll be warm, but not as warm as Florida. It'll also be a hell of a lot less humid. June temps will probably be mid to high twenties in the afternoons, but you might also find its pretty cloudy in the mornings (an effect known locally as June Gloom). Annaheim (where Disneyland is) is a bit inland so the cloud will usually burn off late morning.