You need the key from the old server, or any other server which has the wildcart cert running.
For OS X Server's native services, Server Admin keeps Certs/Keys in /private/etc/certificates.
The key is stored RSA Encrypted.
The passphrase to decrypt is stored in the keychain of that computer.
You'll need to copy the key from the old computer and if you get it from /private/etc/certficates, you will need the passphrase to decrypt it.
You can decrypt a key with
sudo openssl RSA -in /private/etc/certificates/domain.com.key] -out ~/Desktop/domain.com-no-pass.key
you will be asked for the passphrase, you can find this in the system keychain of that computer, you will notice the keyfile has a long string of nunbers/letters, search for that string in keychain, view it and choose Show Password.
If you run any non-apple services (rumps, kerio, etc) the key may be stored somewhere not encrypted.
It may be easier to re-key with godaddy. Godaddy makes rekeying simple, just generate a new CSR on your new server.
NOTE: Any servers running from the old cert/key will no longer be valid if you do this.
Best Practice Tip: Make it a habit of ALWAYS copying the complete package of:
Non-encrypted key
Signed Cert
Intermediate Cert
in a safe place as a zip with the cert name and expire date as the filename (SSL domain.com exp-2013-10-31)
storing the cert without RSA encryption makes is simple to load the cert somewhere else when needed.