E-delivery of statements (and the numerous federal disclosures they contain) can be a requirement for a new account. There are no regulations that specifically allow or disallow this business practice. Depositors who want paper can choose a different account, so nobody is forced into e-delivery.
In spite of your ability to precondition a particular type of account on e-delivery of statements, you still need to get each account holder's "affirmative consent" to e-delivery of statements (because they contain Reg. E and other federal disclosures that must be delivered "in writing.") To be legal, consent can only be obtained in the manner set forth in Section 101(c)(1) of ESIGN.
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...gone fishing.