Each child needs their own valid passport and a document establishing the relationship to the TN principal, usually a birth certificate showing the parent's name. Where names or spellings differ across documents, bring the records that explain the difference rather than relying on an explanation at the counter.
Documents follow the child, not the household
It is common to assemble one family folder and discover at the last moment that one child's birth certificate is a hospital record rather than a civil registration, or that a passport expires mid-stay. Check each child's set independently against the same short list, and where a certificate is issued in another language, have it translated so the officer is not reading around it. Decide who orders what, and put it in writing.
Foreign civil registries answer only to the person entitled to request a record, so a parent cannot always obtain a document naming the other parent, and an employer obtains none of them. Hypothetical example: an arborist preparing a TN transfer assigns each child's folder to one parent, and the parent who has no standing to request a registry copy learns of the restriction in the first week rather than the week of departure. Translation follows the same logic: the certification of accuracy, completeness and translator competence is the translator's to give, and a parent cannot supply it in their place.