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Dear Mr. Medrich
Thank you for posting your request on History Hub!
We are not aware of any federal program that employed children as artists in order to promote the New Deal programs of the Franklin D. Roosevelt administration or that solicited voluntary contributions of art from children to be used as promotional material. We can not categorically state that there were never any examples of private organizations using art by children to promote the goals and ideals of the New Deal, however the National Archives would not necessarily have records related to that.
However, one intersection between New Deal programs and child artists that we are aware of is federal funding of art classes for children at community art centers and art schools as part of the Federal Art Project of the Works Progress Administration. If this is a topic that interests you, there are a number of avenues you can pursue for further research.
Records of the Works Progress Administration can be found in the National Archives’ Record Group 69. Finding aids for this record group can be found in the Guide to Federal Records and the National Archives Catalog. The Franklin D. Roosevelt Library also has at least a few records relating to the Federal Art Project, such as this photograph of an art class for children in at the Walker Art Center. For questions about the records in the FDR Library, you can email email@example.com.
This web page from this Library of Congress also gives information about archival collections relating to the Federal Art Project in the Library of Congress, the National Archives, the Archives of American Art, and the National Gallery of Art. Also, some of the community art centers still exist or have successor organizations that you can contact. Finally, a search online will come up with a variety of articles relating to the project, with just a couple of examples being The "New Deal" Child Artist: Textiles from the Educational Alliance Art School and these essays by FAP employees.
We hope that this information is helpful. Best of luck with your research!