The local authority referred itself to the Information Commissioner's Office (ICO) following what it called an "isolated incident" on September 19 last year.
The council has been given advice on data protection legislation by the ICO, which was updated last year, but will face no penalty from the information watchdog.
In a brief statement, the ICO said: "We have closed this case with no further action. We provided guidance to Wakefield Council on data protection legislation."
Although other public sector organisations, such as the police, are authorised to view details of children in care, it is understood that the incident led to the child being identified by people who should not have legally had access to the information.
Wakefield Council has been contacted for comment.
Local Democracy Reporting Service