This is already something every major US and Canadian bank offers. It's called a merchant account. The API for using a merchant account on the web is called a payment gateway. They all offer them.
Actually no they don't : atleast not in the way paypal or Stripe et al do. I have had meetings with some their "small business advisors" and and I ran screaming to Stripe. Getting and using Interac device and connection was simple enough but their solution for online payment processing will set your hair on fire.
http://www.rbcroyalbank.com/RBC:z7TrmawYUA4BRAAwmCMAAAD8/bus...
http://www.tdcanadatrust.com/products-services/small-busines...
http://www.scotiabank.com/ca/en/0,,557,00.html
https://www.bmoharris.com/us/small-business/services/merchan...
https://www.cibc.com/ca/small-business/merchant-services.htm...