THIS, and please never buy any second hand shoes for them, even if they appear to be exactly the correct size in length and width, as once worn by the original child they will have moulded themselves to the previous child's foot shape.
Also, please don't buy even new children's shoes online, unless they are genuinely refundable, and you personally know how to check that they are not already growing out of them, and that you can tell by feeling them once they are on your child's feet whether the width is correct - too wide or too narrow are both serious problems - you also need to check that the back of the shoe is right, so that the support there is correct, and won't rub or cause blisters!
You have probably experienced yourself that we can try on new shoes in a shoe shop, and even though they are apparently the correct size, they don't feel at all comfortable, which could be for many different reasons.
I know that quality children's shoes are awfully expensive, so if your child has a doting (and reasonably financially comfortable) grandparent or similar, could you ask them to buy your child's shoes until your DC stops growing quite so quickly? We did that for our first Grandchild, but we were lucky enough to live close to a Clarks outlet shop, which helped a bit with the cost.
Apart from maybe their knickers/underpants (parental choice), it doesn't matter what other clothing they wear second or even third hand - as long as they are washed properly, and are still comfortable.
But shoes can, will, and do, misshape a child's tender young bones if not chosen, measured, and fitted, correctly. So I truly believe that in the case of shoes, they need to be bought new, and tried on at source.