Is saying "Let me show you it" totally wrong?

My kids (8-10yrs) love to say things like this. It just rolls naturally out and I correct them often. Is there is a specific reason the grammar is wrong? Maybe for the brain it is more direct than saying "Let me show it to you."

Shall I keep correcting them, or is it OK for them to say it?

I would advise OP not to "correct" the kids' grammar - they've quite correctly recognised the basic principle of how ditransitive verbs work, and contrary to OP's assumption, there's no generally-applicable "grammatical rule" saying you can't use pronouns for both objects in such contexts.

Take for example, demonstrative and indefinite pronouns (this, that, these, those, and some, all, both, each, etc.)...

I'll show him this
You can ask him that
Give me some

...are all perfectly normal English. It might be a little contrived, but grammatically speaking there's nothing wrong with "I'll ask this that" - for example, while pointing to a computer running Google (this) with some particular question in mind (that).