CA sites don't request the private key, simply because if they did, the key wouldn't be "private" any more. When requesting a certificate, usually you upload a certificate request, which requires a private key to be made.
Some issuers offer the possibility to "renew" the certificate, in which case, you don't need a request, because they reuse the fields from your existing certificate.
If you generated the new certificate with a request, then you will need the private key used to make that request. If you didn't need a request, then the certificate probably got issued according to your existing one, so you need to reuse your private key for it.
You can make sure that the key you already have and the certificate matches with the following commands:
openssl x509 -in /the/certificate.pem -noout -modulus
and (assuming you have an RSA key)
openssl rsa -in /the/private/key -noout -modulus
If the two commands give the same output, then the key and the certificate matches.