Geosacy, if you could tell us the area where you currently live (just the city or something as vague as 'North Yorkshire' if you're not comfortable with us knowing too much about where you live) then we can give you the names of various Rescue Centres near you. Many of them are not on rabbitrehome (my local RSPCA rescue centre isn't) so you won't see them on there.
When you know what rescue centres are nearby, you can take your children along with you to see the different kinds they have and discuss with the rescue centre staff which of their rabbits have the right kind of nature for children and they will also tell you what you need to know to take care of one and who should be responsible for the overall care of the rabbit. In some cases they may be able to tell you what health problems some of their rabbits have, but (just like in a pet shop) any healthy rabbit can develop problems later on. There is no such thing as a rabbit who never has to go to a vet.
There are minimum requirements to owning a rabbit these days, its no longer a case of keeping them in a small hutch or indoor cage, so rescue centres will want to make sure you have enough space and can provide a suitable home.
Good luck with finding the perfect bunnies for your family :wave: