"Oni" is masculine personal, "one" is not masculine-personal.
That means, that in "oni" there is at least one masculine human being.
"One" is either a group of only women, or "they" for... well, anything else.
Are you able to just say "Mają dzieci" here? So omitting "One", or is the "One" needed to specify what type of group it is?
Yes. 3rd person pronouns are omitted a lot less often than the others (some would claim that they shouldn't be omitted at all), but especially that in real conversation you'd almost certainly have some context, you can omit them.
"oni" = there is at least one man among 'them'.
"one" = there are no men among 'them' (so almost always 'only women/girls')
Could this be used in a kidnapping situation to state that "they have the children"? If so, it's funny that only context makes the same words mean that they have produced offspring.