You can’t let them take my kids away from me.
I would expect “von mir” here, rather than just the dative alone, “Weg von mir” is "away from me, and “weg” is attached to the verb, so that nails down “away,” but “from” feels missing.
I find the Germans often use dative case where we in English would rely on extra words like “from”.
Sie haben es mir gestohlen = They stole it from me
And sometimes they’ll use dative case where you might not expect to see it at all:
Du hast mir das Leben gerettet = you saved my life
Ich habe mir den Arm gebrochen = I broke my arm
That being said, alternatives are usually available:
Du hast mein Leben gerettet = you saved my life
Ich habe meinen Arm gebrochen = I broke my arm
And, I suspect, “sie haben meine Kinder von mir weggenommen” might also work, if you don’t like the dative variant.
Good explanation of the dative!
“… von mir wegnehmen” does not sound good. “wegnehmen” already includes the “von”.
In this particular sentence you can even ommit the “weg”.
“Du darfst nicht zulassen, dass sie mir meine Kinder nehmen.”
has the same meaning.
But that does not work always:
“Der Krieg hat mir meine Kinder genommen.”
(The war took my children).
means they are dead, and you cannot say “wegnehmen” .