From GoDaddy's Renewing my SSL Certificate docs (emphasis mine):
If you’re using a Domain Validation (DV) certificate with the primary domain for your account, and you’ve set the certificate to auto-renew, no further action is needed on your part. Renewing your SSL certificate is completely automated.
GoDaddy (and other such services) don't have access to customers' private keys, and the only way I can think of this to work is to re-use previous CSRs for a client.
My knowledge in this field is limited, so this is what I base this on:
It's clear that "renewal" is the same as "requesting a new certificate", because SSL certificates cannot be edited (i.e., while also retaining their validity).
A certificate signing request (CSR) is required by a public certificate authority (CA) to issue leaf / subscriber certificates. A CSR is generated from a private key, and if the private key is sufficiently safe, kept secure, etc. then it can be reused to generate a CSR to request a new certificate.
CSRs can be reused, therefore a new certificate can be created simply from a saved one.
If I'm correct in the items above, then this InfoSec SE thread all but confirms my assumption, and this question can be close as a duplicate.
