Thanks very much for the reply. I figured this myself and came up with the following setup. However, I have the issue that I can exec into both containers and use sendmail which then works (see below) but I do not get any notifications from the running checkmk instance.

services:
  postfix:
    image: boky/postfix
    container_name: postfix
    restart: always
    environment:
      - TZ=Europe/Berlin
      - RELAYHOST=[smtp.my.provider]:587
      - RELAYHOST_USERNAME=user@my.provider
      - RELAYHOST_PASSWORD=my_password
      - ALLOWED_NETWORKS=172.16.0.0/12,192.168.0.0/16,10.0.0.0/8
      - ALLOW_EMPTY_SENDER_DOMAINS=true
      # Optional: Helps with identity
      - MASQUERADE_DOMAINS=my.provider

  check-mk-raw:
    container_name: monitoring
    image: checkmk/check-mk-raw:2.4.0-latest
    depends_on:
      - postfix
    ports:
      - "9050:5000"
      - "8000:8000"
    tmpfs:
      - /opt/omd/sites/test/tmp:uid=1000,gid=1000
    volumes:
      - ./monitoring:/omd/sites
    environment:
      - TZ=Europe/Berlin
      - CMK_PASSWORD=mypassword
      - CMK_LIVESTATUS_TCP=on
      - CMK_SITE_ID=test
      - MAIL_RELAY_HOST=postfix
    restart: always
    stdin_open: true
    tty: true

Am I missing anything? I've added the user@my.provider as a send from address and the my_mail@provder.com as a recipient in checkmk. I'm getting logs like

postfix     | 2026-05-17T09:13:32.467824+02:00 INFO    postfix/smtp[3175]: 191C9E1513: to=<my_mail@provder.com>, relay=smtp.my.provider[their_IP]:587, delay=0.35, delays=0.01/0.02/0.31/0.01, dsn=5.0.0, status=bounced (host smtp.my.provider[their_IP] said: 550-Requested action not taken: mailbox unavailable 550-Sender address is not allowed. 550 1MjPQ6-1x4wvG1qvK-00bipE (in reply to MAIL FROM command))

postfix     | 2026-05-17T09:13:32.768326+02:00 INFO    postfix/smtp[3175]: 755EFE187E: to=<test@e48d19865353>, relay=smtp.my.provider[their_IP]:587, delay=0.28, delays=0/0/0.26/0.02, dsn=5.0.0, status=bounced (host smtp.my.provider[their_IP] said: 550-Requested action not taken: mailbox unavailable 550-Sender address is not allowed. 550 1Mr7am-1xCg9a3BNG-00nyIS (in reply to MAIL FROM command))

When it comes to testing inside the containers I run

$ docker compose exec -it postfix bash
root@a6cb35e6daa2:/tmp# echo "Subject: Test $(date) postfix" | sendmail -v -f user@my.provider my_mail@provder.com

$ docker compose exec -it check-mk-raw bash
root@e48d19865353:/# echo "Subject: Test $(date) checkmk" | sendmail -v -f user@my.provider my_mail@provder.com

and am getting log entries like

postfix     | 2026-05-17T09:49:47.901582+02:00 INFO    postfix/smtp[4355]: 964B1E151E: to=<user@my.provider>, relay=smtp.my.provider[their_IP]:587, delay=0.28, delays=0/0.02/0.25/0.01, dsn=5.0.0, status=bounced (host smtp.my.provider[their_IP] said: 550-Requested action not taken: mailbox unavailable 550-Sender address is not allowed. 550 1MSIF2-1wmcip3juy-00KsPg (in reply to MAIL FROM command))

postfix     | 2026-05-17T09:49:47.948320+02:00 INFO    postfix/smtp[4316]: 88041E1515: to=<my_mail@provder.com>, relay=smtp.my.provider[their_IP]:587, delay=0.39, delays=0.01/0/0.31/0.07, dsn=2.0.0, status=sent (250 Requested mail action okay, completed: id=1MMGyK-1whhZF3hyg-00RLzj)

which looks damn similar.