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Hello there, @am23.
Being able to pull up a report with the vendor's account and routing numbers is a great idea in QuickBooks.
While this isn't available at the moment, you can make some arrangement like keeping a hard copy of the data or finding a third-party app to handle vendor payments and give the reports you need.
You may also refer to these articles for additional information about Vendor payment.
If you need further assistance, feel free to comment below. Have a nice day!