I can think of two possible solutions to this:
First possibility: You can set a custom
mailbox_command in Postfix. For each email Postfix receives, it will call the
mailbox_command defined in the configuration file and provide the email as plain text on
stdin. That way, you can decide how you want to process each email. You will have to parse the headers of the email yourself though. You can take a look at the Postfix documentation for more of an idea on how to make this work. Additional applications like
procmail can also act as a
mailbox_command and help you with processing the incoming mails.
Second possibility: Use Dovecot to store emails. You don't need to enable the POP/IMAP modules of Dovecot. However, Dovecot comes with the fantastic
doveadm utility which allows you to query mailboxes and read from them. With
doveadm, you can automatically select new emails, emails from last week, read the whole email, just part of it, delete old messages, and so forth. It's a powerful utility, I use it in various scripts to purge old emails or provide
learn-as-spam folders for users. Take a look at the documentation in the wiki.
(Note: I also posted this on Stackoverflow, where the poster asked the same question)