You cannot. The Apple Card is tied to the Apple ID that was used to apply for it and cannot be transferred.
To continue to use that Apple Card, you need to use the Apple ID it is tied to.
Apple IDs should be maintained updated and accessible at all times, they are not supposed to be changed and forgotten like that because issues can arise. Besides the Apple Card all purchased content is and remains tied to the Apple Id that bought it, and in the case of Apps they will continue to request it for updates.
You'll need to recover the Apple ID, to use the Apple Card, or Apply for a new one. Do note that the old Apple Id will continue to exist and the Apple Card with it, as well as any attached subscriptions and they will continue to charge you unless cancelled.
You can attempt to recover your Apple ID here: https://iforgot.apple.com/appleid#!§ion=appleid