There are literally thousands of youth football leagues across the country which allow children between the ages of 6-12 to play full contact football as well as flag football. There are several large organizations which can help you find a team or start a team in your area. Get onto a search engine such as google and type your city,state,youth football or midget football and hit search. Also, contact your local high school football coach and they can direct you to the closest organization. There is usually a close relationship between the youth and high school teams. Hope this helps. My son's been playing for 5 years and LOVES it. He has become a better student, team player and has a great deal of respect for authority thanks to being involved in organized sports. Good luck!
Try contacting your local parks and recreations councils, your childrens' school district, or ask some of the parents of your childrens' school friends if they know of any little league football organizations in your area. In some areas (like where I live) the little league organizations will sometimes put out placards announcing team sign ups and registrations. My oldest son played little league full contact football last year when he was six. You may also want to check with some of the middle school or high school football and cheerleading coaches to see if they have any ideas on where you can find out if there is a little league football team in your area.
The AYF (American Youth Football) has programs in just about any state in the country. My program here in Chicago is part of AYF, and it is the top tier of pee wee football organizations. They offer better programs and larger opportunities to players that are of all levels, from the kids who "want to try it" to the athelete getting ready to move to the HS level.