This morning, whilst on hold for an hour with Anthem, I poked around on the CoveredCA Facebook page and found some info.
I learned if you have gotten an application # from Anthem (in California, anyway), you can register on their website with your SSN rather than your plan #.
I tried this, and up popped a Member ID # for me, off to the right.
If you try this, be sure to write down the plan # before the system kicks you off.
Others have reported being able to get a temporary ID card
printed, too, but I haven't had luck. I keep getting kicked out.