I would get an "old" horse. NOT a pony. Children at that age aren't ready for the responsiblity of controlling their own horse.
I was in the same place as you a month ago. I had to put my old mare down and the neices have all been on lead line lessons with Soda (spicy young QH gelding). I bought a 13.2 welsh mare that is currently too much for them to handle, but I've also worked with their parents and they will be enrolled in lessons beyond the usual lead line with the gelding. By the time they are ready for the pony she'll be ready for them.
You basically have two "safe" options. 1. Buy an old safe horse they can ride for the next 3-5 yrs, it may be to big for them but it'll be safer. 2. Buy an old small pony. Your 4 yo will grow out of it in 2 yrs but you'll get more time out of the younger child.
Best option is lessons along with an older horse they can "ride" out with the family.