For code signing an iOS app, you need 3 things:
- iOS development or distribution certificate
- iOS development or
distribution provisioning profile (that is set up for use with the
cert above)
- private key (the private key goes in your key chain and
was used to generate the certificate above).
Apple does not store the private key, not should they. That is something you need to keep and protect as it allows you to prove that you are the rightful developer who can sign an app as you. Do not share the private key.
If you lose the private key, the only way to sign an iOS application is to create a new new private key, generate a new certificate and provisioning profile, and use those to sign your app.
If you can still log into the old Mac, you should be able to go into the Keychain Access app and find the certificate (and associated private key) from your old account and import them on the new machine.
If you do not have an iOS role that allows you to create the certificate and provisioning profile, you will need to get the person who has that access to create those for you, and then send you the privite key. You can download the cert and profile from Apple's dev site after they have created them.