Archive.org hosts a wide array of downloadable material, including movies that have entered the public domain. The site doesn't have a category for children's movies, but the animation section has a number of movies appropriate for kids.
Many of the films on Archive.org are from the 1950s or earlier. In the first half of the 20th century, film studios often neglected to renew the copyrights on their movies, so they entered the public domain, making it legal for websites to distribute them for free. However, parents might want to preview cartoons before sharing them with children, as Archive.org doesn't rate films, and many cartoons from the 1950s and earlier contain racist content and other objectionable material.