I would like to export some simple vector animations created with After Effects, and keep the animation in a vector form, such as the Flash format. I am using After Effects CC and it seems that there is no built-in export procedure to do so. Is there a way to do this?
You can export to the vector format SVG using an AfterEffects plugin called Bodymovin. It has some limitations, but is a very useful plugin.
If you really want to export to Flash, export to SWF format. Though you may run into some issues doing that, so looking at a video tutorial may help.