When someone opens an account and receives a Reg E disclosure, is it required that they be provided another disclosure for any subsequent accounts they may open, if nothing has changed?

No - see Staff Commentary 205.7(a)-3:

"If a consumer opens a new account permitting EFTs at a financial institution, and the customer has already received Regulation E disclosures for another account at that financial institution, the financial institution need only disclose terms and conditions that differ from those previously given."

We use DepositPro, which is set up to print the current Reg E disclosures on all new accounts - CSRs can't change the disclosures. Who would be responsible for ensuring that the disclosures are given out when needed?

If the customer wants wants a new service, there is a "close proximity rule". From Reg E commentary 205.7 comment 1

On the other hand, if an agreement for EFT services to be provided by an account-holding institution is directly between the consumer and the account-holding institution, disclosures must be given in close proximity to the event requiring disclosure, for example, when the consumer contracts for a new service.