The Site Admin API does include an ExportProject method.
Make note there are limitations on the Project Export functionality. There is a limit to the amount of data that can be exported. Refer to the limitations information provided for the feature in the Administrator Guide.
Also, the project has to be deactivated during the export. The API document doesn't indicate if the method will do that automatically, or if you would have to also use the DeactivateProject method and the ActivateProject method.
Make note there are limitations on the Project Export functionality. There is a limit to the amount of data that can be exported. Refer to the limitations information provided for the feature in the Administrator Guide.
Also, the project has to be deactivated during the export. The API document doesn't indicate if the method will do that automatically, or if you would have to also use the DeactivateProject method and the ActivateProject method.