Yes, children can have Achilles Tondonitis, however, it is rare. The Achilles tenodon is the longest tendon in the body starting at the heel and connecting all the way to calf. Children more commonly have a condition called severs disease. Usually children experience severs disease from overuse, a biological condition, and "growing pains".
It is always best to get yoour children looked at by a podiatrist whenever they are experiencing pain in their feet. Treating children's problems eary will decrease the risk of the problem getting worse or from the problem causing other and worse problems.