allows my mail clients to connect via IMAP to view and search emails
dovecot will be able to handle this part. This is what I use as a mail archive (once a year, archive all mail from the previous year from various mailboxes to my self-hosted dovecot instance). I wrote this ansible role for it.
downloads new emails via IMAP
As others recommended, imapsync should be able to handle that part.
docker solution
These tools are simple enough to install and manage (one package, one config file), Docker is not needed. If you really need it to fit into your docker-based setup, build and maintain your own images.