From the Feet Up
One of the problems I come across a lot as an Enjoy-a-ball coach is ill fitting shoes. When we play running games I am constantly watching the children and assessing different things. First thing I check is the child's shoes if I have any concerns on their development and 9 times out of 10 that's why I'm noticing something "not quite right"
The wrong sized shoes can cause many issues. From blisters, pressure sores and ingrowing toe nails in the short-term, to feet deformities like hammer toe and knee and posture problems in the long-term. Shoes that are too tight, too loose, or without enough support, can lead to unwanted stress on the feet, ankles, lower leg, hip, and spine. This ongoing pressure can lead to some major issues.
Your child is at a key time in their development. If their shoes do not fit properly then often they develop a running style that keeps the shoe on their feet. A running style that they could have for the rest of their life.
I understand that money is a major issue in this. I'm still growing into the size 11 boots my mum bought me when I was 15. A fingers worth extra room should last a while. Any more is too big.
When buying shoes please take your child to a shoe shop, "Clarks" is great. (I'm sure others are too) Their feet are measured properly and the quality really good. If you don't want to buy there that's absolutely fine but at least you have the size. There are some places I wouldn't buy gym rubbers from as the really cheap ones don't last long at all. This develops other complications, the fabric at the back and sides wears quickly causing the childs foot to come out or it doesn't have the support required.
I don't personally like gym rubbers but apparently they allow the child's ligaments and muscles around the ankle to develop. That's important for stability. Please please please have a look at the gym rubbers before you buy them. If they don't look like they should be used for running about in then they probably shouldn't be used for gym or physical activity. The best ones for me are the ones without laces and without Velcro. Makes things easier for everyone.
This is only my opinion, and with a bit of research I've written this long post. Disregard it if you want but please consider what I've written. I don't have any children and would never tell anyone what to do with theirs. I consider myself reasonably informed in such a topic and only want to help your child to develop.