I doubt it. Rolling out something equivalent to a national ID system greatly complicates the already difficult logistical challenge of getting up to 3 vaccines requiring more than one shot to 70% of the population. And then even if we could do that, people would have to constantly present that card. If the gate agent checks their card, then some anti-masker would simply take off their mask in the jet bridge. So, then the FAs have to check. Are the FAs supposed to remember the status of all 100-200 passengers on every flight? And contrary to our egos, pilots are nothing special. We don't get to flash a card at the hotel and expect everyone there to treat us differently.
Even once the vaccines are distributed, we will be wearing masks until we reach herd immunity. If everything goes to plan, that's expected to happen in May-June.