You won't be able to take a snapshot of a current box and use it as a docker container, but you can certainly create a container to use on your EC2 instances.
- Create a Dockerfile that builds the system exactly as you want it.
- Once you've created the perfect Dockerfile, export a container to a tarball
- Upload the tarball to S3
- On your EC2 instances, download the tarball and import it as a Docker container.
Are you planning to use something like s3fs to mount an S3 bucket? Otherwise you can just copy the tarball from your bucket either as a userdata boot script or during a chef/puppet/ansible provisioning step. Depends how you want to structure it.