I think imapsync will fulfill your needs.

https://github.com/docker-mailserver/docker-mailserver should be able to do it.

if i remember this in 6 weeks i’ll check the setup at work.

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.

Piggybacking on this question: any solution that provides really good indexing on those local mails, for fast wildcard searching?

I use dovecot for this. And thunderbird to actually move/archive the emails. I use caddy for many of my services, so I have pointed dovecot to caddys certificates (for “my.domain”), since it manages certificates through let’s encrypt. I had a plan to install postfix for sending internal emails from my self-hosted services, but it seemed like a bit of configuration and I got busy with other stuff

I made an excerpt from my docker-compose.yml, but you probably have to figure out some things on your own

version: '3.4'

services:
  dovecot:
    image: dovecot/dovecot:2.3.20
    restart: unless-stopped
    volumes:
      - ./dovecot/:/etc/dovecot
      - /mnt/storage/dovecot/mail:/srv/mail
      - ./caddy/data/caddy/certificates/acme-v02.api.letsencrypt.org-directory/wildcard_.my.domain/wildcard_.my.domain.crt:/etc/ssl/cert.crt
      - ./caddy/data/caddy/certificates/acme-v02.api.letsencrypt.org-directory/wildcard_.my.domain/wildcard_.my.domain.key:/etc/ssl/key.key
    ports:
      - 993:993

contents of ./dovecot folder:

dovecot.conf
passwords

contents of dovecot.conf (I think I searched online to find a good example, I don’t remember where from…)

## manage this file

mail_home=/srv/mail/%Lu
mail_location=sdbox:~/Mail
mail_uid=1000
mail_gid=1000

protocols = imap pop3 submission sieve lmtp

first_valid_uid = 1000
last_valid_uid = 1000

passdb {
  driver = passwd-file
  args = scheme=argon2i /etc/dovecot/passwords
}

ssl=yes
ssl_cert=</etc/ssl/cert.crt
ssl_key=</etc/ssl/key.key

namespace {
  inbox = yes
  separator = /

  mailbox Drafts {
    auto = subscribe
    special_use = \Drafts
  }
  mailbox Sent {
    auto = subscribe
    special_use = \Sent
  }
  mailbox Spam {
    auto = subscribe
    special_use = \Junk
  }
  mailbox Trash {
    auto = subscribe
    special_use = \Trash
  }
  mailbox Archive {
    auto = subscribe
    special_use = \Archive
  }
}

service lmtp {
  inet_listener {
    port = 24
  }
}

listen = *

log_path=/dev/stdout
info_log_path=/dev/stdout
debug_log_path=/dev/stdout

Paperless-ngx might be worth a look. Probably overkill if you just want to store mails but great for general purpose document management.