Looking for an easy way to connect to all major email apps

Is there any API service that allows me to integrate with all major email marketing services(like Mailchimp, Constant contact, sendgrid, etc) and use the same API calls, like a wrapper?

For example I want to allow users to connect with their email service and schedule a campaign. Since there are so many email services, it would take some time to integrate each separately, it would be awesome if with one service to integrate all of them and use a general API for all.

I’m not entirely sure which angle you’re coming from, but https://segment.com/ does a good amount of those but not all.

Yes, something like segment.io but for email services. For email they only have mailchimp which I need.
I’ll need getresponse, aweber, constant contact, iContact.

How startups like promoter.io handle the email delivery? They send the emails from the client SMTP or they use the same domain for every email, just with a different reply-to email?

Look into integration tools such as Zapier or Cloudwork, which I listed in this entry:
http://www.oliviertravers.com/archives/2015/03/11/saas-integration-trends-players/