Does an online credit card app need to comply with E-SIGN? If so, does it comply if the card applicant completes the app online with a bank provided computer/device?

"Compliance with ESIGN" is only necessary when you want to deliver electronic documents and count them as the legal equivalent of paper documents. Primarily, the need to assure equivalence arises when documents contain disclosures that a federal law or regulation (Regulations B or Z, for example) require you to deliver "in writing."

To determine whether or not ESIGN is applicable in your case, make a list of all forms, notices, disclosures, and other documents you are required to present to each credit card applicant in writing. If you used to deliver them on paper and now want to use some type of electronic alternative, you must obtain each applicant's consent in the ESIGN-prescribed manner.

And if E-SIGN is applicable, I'm assuming that a customer completing an app on a bank computer is not demonstrating the customer has the hardware and software necessary to access the documents in question because a bank computer is not his/her personal computer or device. correct?

That's the way I see it...except I'd draw the line at "controlled access," not "ownership." Customers might rent or borrow their hardware from third parties. Unless the bank is in the hardware rental business, customers have no guarantee that the bank's courtesy machines will be available indefinitely.

For example, most public libraries today offer online access using library-owned computers. A customer could use those computers for all the consumer's online access. If that access included the ability to print documents, the consumer could provide demonstrable consent for e-delivery of written notices, disclosures, etc., via the library's machines.

Our credit card program operates as I described. The vendor then mails the required docs to the applicant. This seems like a work around to the demonstrable consent.

Will mailed paper documents satisfy the timing requirements for any disclosures you must deliver before or during the application process? (It's been a long time since I was familiar with these rules, so the best I can do is help you identify possible problems.)