If you want to generate certificates, you need to cause the client to generate a key pair, and send you at least the public key. You can do this in Firefox via a Javascript call, it's crypto.generateCRMFRequest. I'm guessing there are browser-specific methods available in other browsers too. But first, you need to figure out how to issue a certificate once you get a public key.
You could script something on the server with OpenSSL, but it has built-in support for CSRs, not the CRMF format Firefox will send you. So you'd need to write some code to convert the CRMF to a CSR, which will require some sort of DER processing capability… I'm just scratching the surface here—operating a CA, even for a toy application, is not trivial.
SSO solutions like OpenId and PKI solutions do overlap, and there is an elegance in PKI. But the devil is in the details, and there are good reasons why this approach has been around a long time but has only taken off in government and military applications.
If you are interested in pursuing this, follow up with some questions specific to the platform you would want to develop your CA service on.