The chemicals definetley can cause damage. Depending on the type of perm you use and how long it's on the hair and such things. I would suggest taking the child to a professional if you really want their hair permed. They may advise agaist it. I would, and I probably wouldnt do it.
People will have all different opinions on this one, I was a child who had gotten a perm. I was horrified afterward. I had straight hair since birth. I got my perm at 12, I have not had straight hair since. I'm 30.