The US government did not require that shipping lines or ships' captains list passengers by name until 1820.
On Jstor there is an article about the migration: "The 1809 immigration of Saint-Dominque Refugees to New Orleans: Reception, Integration & Impact" by Paul Lachance
On page 111 footnote #8 it states where lists of passengers keep by the mayor's office could be found. One cannot copy from the articles on Jstor, so you will have to look at the citation in the article.
Jstor is normally only available at libraries or universities, but during the pandemic they are allowing people, who get a user name and password, to view 100 articles for free from home.
Dear Ms. LeGardeur,
Thank you for posting your request on History Hub!
Even though there are no NARA passenger lists into New Orleans prior to 1820, we searched the National Archives Catalog and located 18 series in the Records of the U.S. Customs Service (Record Group 36) that were created by the Collection District of New Orleans (Louisiana). 1804-1913 and may include references to inbound passengers in 1809. For access to these records, please contact the National Archives at Fort Worth (RM-FW) via email at firstname.lastname@example.org.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RM-FW. We apologize for this inconvenience and appreciate your understanding and patience.
Ancestry.com has a searchable database, U.S. and Canada, Passenger and Immigration Lists Index, 1500s-1900s, that may contain the New Orleans passenger information you seek. There may be a fee for using Ancestry. Instead, check your local library. Many library systems subscribe to these sites, making them free for their patrons.
We hope this is helpful. Best of luck with your research!