None of the readers with WiFi can access the Internet unless you set it up with your W-LAN password, so that shouldn't be a problem. I'm not aware of any somewhat current readers or tablets without W-LAN.
How young is the child we're talking about? E-ink readers are sadly rather fragile (the screen can easily crack under pressure or torsion) so for a young child a (cheap) tablet might be a better choice.
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