On the east coast, there are a lot of open tournaments this (every) summer:
- Continental Class
- World Open
- Manhattan Open
- Atlantic Open
- Continental Open
- Potomac Open
- Northeast Open
- Bradley Open
If your child is advanced player, you could consider:
- Barber/Denker/NGIT/US Open
- Susan Polgar Foundation Girls Invitationals
- Cadet
- US Junior Closed
- US Girls Junior Closed
- North American Youth Chess Championship
- North American Junior Chess Championship
- Pan American Youth Chess Festival
- Philadelphia International
- New York International
- Washington International
And summer ends with Labor Day, on which day almost all states have their own state opens.
In addition, you have weekly club tournaments, and chess camps. So it's a great time for your child to make big progress.