Supporters argue the change could bring in more funds, while opponents worry the city would lose control over collections.
NOLA Public Schools will host a virtual town hall for parents with vaccine questions on Wednesday evening.
Three selective elementary schools will still have additional requirements after students apply through the centralized portal.
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.