Parsing emails to database

I have an idea of creating service for our customers where their automated emails from certain services would be opened and parsed and that information than written to our database. One idea is to keep customer emails as default on those sites and then redirect emails from certain addresses to our mail where it would be parsed and other is to change customers default email to address from our mail hosting and then (if they want) just forward them to their private mail. Is this doable and what would be the best way?

Regarding the technical part, what is the best way to create a script that would check if there are new emails, recognize correct mail format and then parse them in order to add information to the database? Should that script be periodically repeated to check if there are new emails or is there a way for a mail client to notify our script each time it receives new mail?

Thank you very much for your answer in advance