How does use of Budget Feeder comply with my bank's terms
When you accept the Budget Feeder Terms, you grant a Limited Power of Attorney specifically for the purpose of accessing your account to sync transactions. This is the basis under which Budget Feeder is able to access accounts.
Budget Feeder makes use of the same type of Bank Feeds that YNAB also use for the bank feeds available to their customers in the USA.
Budget Feeder has also been vetted and accepted by YNAB for access to the YNAB OAuth API.