I don't think that feet need support - the key thing is cushioning - soft springy soles or added insoles.
If they are not expensive then two pairs, one to stay at school for wet days, if that is possible - perhaps a spare pair of socks in a jacket pocket.
I have noticed that many school shoes are very badly shaped with narrow toes and stiff heavy soles - they are very bad for children's feet causing deformities of the feet and ankles, even affecting the knees. Footwear should be light and flexible - like plimsolls. The lace up or elasticated ones seem best. I was teaching maypole and morris dancing at a school recently and the Velcro fastening ones were coming off and children were tripping up