Did you know that most races allow kids to run for free (ages 8 and under)? These races are known as "Kiddie Races." If you are interested in having your child participate for a race, ask if there is a free kiddie race. If not, sometimes, they will allow kids to run for free if you tell them that you are not interested in purchasing the race T-shirt.
Do a internet search using the key words "Road Runners" or "Kids' Races" and include your local town or city. This will bring up the most popular sites for your location.