I think all kids should have rain pants in this area! It's wet and muddy for so much of the year that it would be great if schools could put on rain gear and send them outside.
Either something like the one piece rain suit or the separate pants that are worn with a jacket (suit would be faster, but separates may get more use) -
https://oaki.com/collections/boy-rainwear
OP I would talk to the school. Is the idea that they'll be indoors and outdoors both or only outdoors? If both, I would think they would want rain boots that can be changed into "indoor shoes" so that kids aren't tracking mud all over the school. If you go with boots, try to find ones that are more fitted at the shin so water doesn't get into the boot.
For regular footwear, I really like Doc Martens. We wear them for day to day use from fall to spring. They're warm enough for winter, water proof enough for day to day slushy conditions, and sturdy.