The following is a list of some of the best-know and most reputable summer gymnastics camps in the US. I've compiled some of the basic information by looking at the camps' respective websites. Seek out those sites for more information.
International Gymnastics Camp (IGC) -
Location: Pennsylvania
Ages: 7-17 (co-ed)
This camp offers week-long sessions throughout the summer for gymnasts beginner to advanced. They, like most camps, offer other general camp activities like a climbing wall. They've got a great staff of elite coaches, National champs, and Olympic gymnasts. This summer they've had a lot of Olympian visitors, including Nastia Liukin, Shawn Johnson, Chellsie Memmel, Paul Hamm, etc. The facilities are very conducive to learning; they have five different gyms - an Olympic gym, a beam gym, an uneven bars gym, a vault gym, and a dance gym. As far as training facility goes, I think ICG wins.
Lake Owen Camp -
Location: Wisconsin
Levels: Boys and girls, levels 1-10 and elite.
Lake Owen is a fun place with a lot of great coaches. They have clinicians who are All Americans, National Champions, International Competitors, Judges and Olympians, and they have a lot of college coaches there as well. Like IGC, Lake Owen has Olympian visitors, and in the past have had Carly Patterson, Dominique Moceanu, Shannon Miller, and Kim Zmeskal. They have a floor gym which includes beams, pits, and vault tables, and they also have a separate bar gym. All their facilities have the best equipment available. At the end of the week, this camp has an exhibition where parents and friends can watch what the gymnasts have learned over the course of the camp session. One of the unique things about this camp is that they have really fun recreational activities. They have one of those giant blob that you can bounce on into the lake, as well as tubing, skiing, and a ropes course.
Camp Woodward -
Location: Pennsylvania
Co-ed. Gymnasts are grouped by age and ability.
This camp has special guests each summer, including US Olympians, International Olympic medalists, National teams from many countries, and coaches from the US National team and many Universities. This summer they've been visited by Bridget Sloan, Sam Peszek, Svetlana Boguinskia, Courtney Kupets, Courtney McCool, and tons more. The facility includes a beam gym, a pit gym, and a fully-equipped open gym that's available for 4.5 extra hours of practice for kids who want a little extra gym time. They also have a separate men's gym. There is also a Woodward West located in California that opened in 2003. It also has a gymnastics camp for kids 7-17.
Karolyi Camp -
Location: Texas
Ages: 7+
This camp was established by and is directed by Bela and Marta Karolyi. This camp actually offers some of the most reasonable rates, as far as gymnastics camps go - about half the cost of the others, which is a definite plus. They have a 25,000 sq. ft. gymnastics complex. Not much information is available online about the equipment but I would assume that the Karolyis provide nothing but the best, especially considering this is the USA National team's training center. Other than gymnastics, a bunch of other activities are available. They have an Olympic sized swimming pool, a petting zoo, and a lake for boating, as well as horseback riding, basketball, tennis, and other activities.
There are, of course, many smaller camps in different states like YMCA camps, so if none of these camps are in your area or if you're not willing to travel, a Google search should be good to help you find something that's more convenient for you. These are just some of the most notable camps in the country. I hope you find the right summer camp for you or your kids!