If you aren't able to log into Apple ID in your child's iOS Settings or find that it is greyed out it is likely the iOS restrictions on your child's device need to be disabled.
To disable these restrictions on your child's device, please first perform a software update to your child's device:
Settings > General > Software Update
NOTE: If no update is available, please hard reset your child's device by holding down the power/home buttons at the same time, until the device restarts
Then, please navigate to:
Settings > Screen Time > Content & Privacy Restrictions > Enter Code / Create Code (if restrictions were previously enabled on the device, these restrictions will carry over) > Toggle ON Content & Privacy Restrictions > Under ‘Allow Changes’ section, toggle ON Account Changes
Once you've performed the steps listed above, please try again to log into Apple ID from the iOS Settings.
If you find the Apple ID area on your child's device is still greyed out, preventing you or your child from logging in, please contact our Support Team directly at firstname.lastname@example.org along with any helpful details regarding this issue.