Hi and welcome to YT
I have a four year old and a one year old and they have been great with our yorkie Twinkle. We picked a larger puppy which I would recommend - she's 6months old now and 8lb, so still a very very small dog but a little more robust than if she were 3 or 4 pounds. I would advise, as others have, that the children sit on the floor and let the puppy come to them rather than following it around, and of course always with adult supervision. I wish you lots of luck with finding the yorkie of your dreams very soon
xxx xxx