Pony Club. They're as scarce as hens teeth because families keep them on as pets. (I've still got mine...). But they do the rounds of the local PC Branch bouncing from one to another as the child grows. Call your local PC DC and speak to them. They don't tend to get advertised but pass by word of mouth. Also be prepared for a lot of questions about where and how it will be kept. The good ones (and in fairness the bad ones too!!) tend to be very much loved and with a lot of competition to take them the owners can afford to be choosy.
That said... my daughter started her riding career aged 4 as a show jockey for a local producer. (She could already ride - in as much as a child that age does.) We didn't get our own first ridden, who came from the pony club, until she was 6. So if you have a good local showing producer and the child is a competent rider than that too is a possible route!
As Shay says, PC is probably the best place. I think most ponies go via word of mouth, that's what I found anyway even from a buying point of view. I struggled to find one suitable for sale that wasn't a veteran, I think people hold onto good lead-reins for a long long time lending them out to friends etc.!