I got Vigilante, the dog, and I brought it home to Hjerim, the Windhelm house and my 2 kids won't ask to keep it. I have no other pets and I have left the house and came back 3 times and they won't ask. Is there a console command? If so, please let me know.
This might be a very big bug for those who want dogs. Try to wait a few days to reset some dialogues, or reload a save. You mentioned your kid has a fox. You can't have a dog to be kept in your house unless they don't have pets. However if you adopted the fox after the bug you encountered, try to download the Unofficial Patch.
You mentioned in a comment that your children already have a pet fox. This is probably preventing the dialogue asking to keep your dog as a pet. After all, if they have a pet, they aren't desperate to have one. Also, although the descriptions of adopted child interactions suggest that children will always ask to keep your dog, it might be randomly selected from the child interactions available. You might have been unlucky.