You do not need to have any DMV paperwork with you to get a smog check. The smog technician can simply enter the required information, your vehicle's vehicle identification number (VIN) and license plate number, right into the smog machine. Having the DMV renewal notice simplifies the process for the smog technician as he or she would be able to scan the bar code on the DMV document which would automatically enter the required information into the smog machine.
A very few smog stations have been known to charge extra for their time to enter the vehicle information manually, however this is vary rare. But to answer your question; no paperwork is required to get a smog inspection in California as of this time.
And in case, for any reason, if the DMV website won't allow you to renew your registration online through them, you may want to use a third-party registration service provider like www.NeedTags.com
to get it done without having to go into the DMV to get your registration renewed.