No, you must have your card.
7/6/2015: Applied and paid state fee online
7/7/2015: Went to local PD to get fingerprinted (paper) and pay local fee
7/8/2015: Dropped prints in mail to ISP
8/6/2015: Found this thread, emailed ISP to inquire
8/7/2015: Called local PD (couldn't really tell me anything), few minutes later got an email response from ISP saying they were running background check now and would be approved today if clean, few hours later it shows as approved on the website
Don't know if it was just a coincidence, or if emailing and calling prompted them to process me. Also, seems like paper prints do indeed take much longer than digital.
Does anyone know if I can start carrying now as long as I have my permit # on me? Or do I need to wait for physical card?