What can I do if I have completed a CSR for "domainA.com" but later want it to also apply to "domainB.com"?
I'm assuming there aren't many options once the CA has issued the license, apart from buying an additional SSL cert?
In the vast majority of circumstances, that is correct.
Depending on the issuer, there's a minute chance that they will let you revoke the old cert and then re-issue on the new domain, but you ought not plan on that being an option.
For a single certificate to apply to multiple domain names, it needs to make use of the Subject Alternative Names field, which will need to be set up at the time that the certificate is generated.
The certificate authorities tend to charge a premium for this, in most cases you'll be better off just getting a second certificate with the other domain name (and a second IP address bound to the server to put it on).