Does Postmark have a POP3/IMAP server?

Since Postmark is not a mailbox provider there's not the ability to generate mailboxes for receiving email using IMAP or POP3.

Postmark does offer inbound processing for receiving messages. Inbound processing allows you to send inbound emails (hash@inbound.postmarkapp.com) to Postmark, which we then process and deliver to you via a webhook in nicely formatted JSON. This tool is useful if you allow people to send or reply to your outbound emails (comments, tickets, etc), which can then be processed by your application. For more details, please see our developer docs.