First, that would be immensely subject to fraud and falsification, to the point of being virtually meaningless. Unless we're talking about employers and businesses being able to connect to a central, secured registry of some sort, which crosses TONS of privacy lines, we're talking about something that anyone with rudimentary photoshop skills could falsify. That's a big part of the reason the "immunity passport" idea has been a non-starter here. Maybe it could happen in countries with more authoritarian governments or weaker privacy laws, but it wouldn't be practical here.
Second, masks are more effective at protecting others than protecting yourself, so someone standing next to you without one is actually putting you at risk. If you're vaccinated, an unvaccinated person standing next to you poses no risk to your health. We have all but eliminated many, many once-common diseases without making vaccinations truly mandatory, and some of those have much higher R0 numbers and therefore require a much higher percentage of the population to be vaccinated to provide herd immunity than it will take with COVID19. If no one has been demanding proof of your measles vaccination (R0 around 18, mortality around 15%) to enter their business, they're not going to do so for COVID19 (R0 around 3, mortality around 1%).