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As sp332 asked, IMAP, aka the protocol that 99% of mail clients use for reading mail, is going to be your best answer. You can find any number of imap libraries for your language of choice, from there its a matter of connecting your mailbox and doing what you need pragmatically. Youll be unlikely to find an actual REST API mail provider, because its an unnecessary layer to an existing "API"

I was actually building something like this earlier this year. Depending on your email provider, they will have an API, eg Google's Gmail API, Microsoft's Graph API, etc.

Do you mean like IMAP?

Ask HN: API-Enabled Email?