You can use plaid.com for that. It also integrates with stripe.com pretty well too.
Ideally building from scratch because the platform relys on taking a small percentage of each transaction for revenue.
Plaid seems like an incredible company, going this route would save a huge amount of engineering initially, but would cut into revenues. I might explore this further. Thanks!
If you're in the EU, read up on PSD2. It's opening up payments significantly.
US Based, but I read an article about it. Pretty interesting.