Dryades runs Singleton Charter School, which is in the final year of its operating contract and could close next summer.
State officials say monitoring compliance is up to local school district authorities.
School officials expect the FDA to approve the COVID-19 vaccine for 5- to 11-year-old children in coming weeks.
The group runs Singleton Charter School and was granted a restraining order that prevented the district from closing the school over the summer.
The resolution was introduced by J.C. Romero, who grew up in a non-English speaking household in New Orleans.
The board also introduced a new anti-retaliation policy to protect parents and school staff.
The warning letter, an elevation from a previous and less serious warning that was not posted publicly, says the school must complete a review of all special education files by early next year.
COVID-19 cases in district schools remain low.
NOLA Public Schools reports Singleton officials said the employee has been terminated.
The NOLA Public Schools district recently criticized a state policy loosening school quarantine guidelines for unvaccinated contacts.