I was a gymnast with Atlanta School of Gymnastics from 1972 through 1981, then I started teaching there. I coached high school gymnastics for a year in PA and have been the Head Coach at New Milford High in CT for 20 years.
Yes, many High Schools in the USA have gymnastics teams. The level of commitment is significantly less than club gymnastics, as most school teams only practice a couple hours a day and only during the season. Most of the girls on my HS's team either play other sports in fall and spring season, or they train at a club and "take time off" from that to compete for our school.
There are a couple of high school gymnasts on Chalkbucket.
Im from Ohio and we have many high school teams in the northeastern region. Though now I think they have strict rules that don't let girls compete in both club and school which has discouraged many high school teams.