Thermals, there are some makes that are good enough to wear for a whole week without washing.
Fleece
Socks, long unribbed proper ski socks to be comfortable in ski boots (extras)
Outer water/snow proof layer: For younger children a snow suit maybe best, but they do grow out of them quicker and it is harder to get out of for bathroom breaks
or
A good jacket, which can double up as their winter coat
Ski trousers
Snow boots
Waterproof gloves, with long cuffs to cover sleeves, or mittens. Extra pairs
Indoor shoes/slippers
Hats Extras
Swimsuit
Labelled bag for children going to Kindergarten
Small rucksack/bum-bag for older children, they might not like carrying it, but it a way to ensure they have somewhere to store their own water, snacks and if they need to, keep their spare layers if it gets warm. Especially useful if they are off on their own or taking lessons.
Pocket Money to allow them to buy themselves a hot chocolate if they are out on a lesson
Snacks, to give them a boost when out on the slopes and keep them going till lunch and dinner.
Sun cream
Lip balm
Sunglasses and or Goggles, goggles are better with helmets and are work for all conditions.
Disposable hand and foot warmers, invaluable if children develop cold hands or feet, means you won’t have turn around and go home but can continue enjoying your skiing day.
Small first aid kit (thermometer, antiseptic cream/wipes, tweezers, plasters)
Paracetamol and Ibuprofen
After-sun
Health record of your children if they are going to Kindergarten
EU health cards if travelling in the EU