I'd like to build an email drop service. Is it anyhow possible to receive incoming emails with Google cloud functions, process them and store them in firebase db?
I'm thinking of something similar to Amazon simple email service SES in combination with lambda functions. Does any of the available email services like sendgrid, mailgun, postmark or any other has api's to trigger Google Cloud Functions on incoming emails?
Yes, services like Sendgrid and Mailgun allow you to set up webhooks for incoming mail. You can configure these webhooks to point to an HTTPS Cloud Function, at which point it will be invoked each time you get a new incoming email.