On our website, we offer credit cards with our names on it, but we do not service them. I know we need a speed bump letting the customer know that they are leaving our site, but should a disclosure also be on the link to let them know our Bank does not service the cards?

There is no need for such a disclosure as the card application/agreement will have language that says, "This card is issued by [Bank Name]"

Of greater importance is your third party risk management monitoring the issuer to ensure that they are complying with applicable provisions of Reg Z, FCRA, FDCPA, complaint monitoring, UDAP, etc.