Civi has integrations for GMail and Outlook 365, but also has a feature Pete mentions in the comments called "Inbound Mail". This works with all mail systems, and is also more seamless than the integrations.
The official integrations are for having LESS automation - e.g. if two people have the same email address, it will pop up a box to let you choose, while the automated method picks the first matching contact it finds. Or let you use CiviCRM as an address book. You may still want them.
For what you want, I would set it up like this:
- Create a new mailbox at your organization, e.g. "[email protected]".
- Set a filter in your mail program to automatically forward (or better, redirect) a copy of each email to this new mailbox. Note you can tweak your filter to exclude certain types of mail (e.g. listservs).
- Configure CiviCRM to check this mailbox (Administer menu >> CiviMail >> Mail Accounts, then Add Mail Account). Exact directions are in the documentation linked by Pete. Also be sure to read about permissions for users to edit inbound emails - this is new in Civi 5.31+.
- Make sure the Process Inbound Mail job is turned on in Scheduled Jobs (Administer menu >> System Settings >> Scheduled Jobs).
- If you need further customization, it can be done through extensions. E.g. this extension was written for an organization that wanted two inbound email addresses, and emails sent to the second address were restricted to privileged users.