USBAnk seems to have an API for this, I expect this to be a REST service, but you'll need to create an account to view the documentation. Might be worth a shot and implementing this with a REST GET or POST method is quite straightforward.
See https://developer.usbank.com/products/account-validation/v1
The one I encounter most often is Plaid (plaid.com) Its a platform built for the kind of use case you describe. I don't see a module in the marketplace, but it is an API first platform, so integration should be straightforward. It does not use a routing and account number directly, but pops up a window where the end user authenticates to their bank account and then passes back tokenized authentication information. This eliminates risk that you would incur by having ach and account number information inside of your Mendix app.