With so much uncertainty I don't see how government can compel people to take a Covid vaccine and go so far as to embed vaccination information in ID cards.
I would be shocked if COVID-19 vaccine information would ever be required to be stored on a standard ID card issued by a state or federal government in the United States. It took nearly 15 years to get REAL ID started in all 50 states, and the underlying stated rationale for it was far more bipartisan (protection against terrorism) than the response to the COVID-19 pandemic has been. I'd be surprised if there's even a universal standard rolled out for optional inclusion of said information on ID. There's basically no way to put it on an existing ID that'd be even remotely verifiable (maybe on the magstripe, but that's not human-readable and adding a sticker would almost certainly be trivial to fake.)
All of that is even before considering the decentralized nature of the rollout of the vaccine, and the fact that there's no universal photo ID for every American. The vaccination distribution will be handled by state and local departments of health, in partnership with local clinics and pharmacies. There's currently no national database that I'm aware of that stores who got which specific vaccine, especially to a level where you could definitively prove that this particular person got this particular vaccine. I wouldn't be surprised to see some sort of app or certificate for vaccination available to show voluntarily, but something that's federally-driven seems unlikely given that vaccinations will likely start within the next month or so, and there's no current platform set up for that information.
Same on the international stage. We might eventually see some sort of vaccination "visa" in passports, but as far as I'm aware that doesn't exist in a universal, stored-in-the-passport standard, and any new standard will take time to develop. What's more likely is that you'll see vaccination records required to be issued a visa, and possibly some tightening on who needs to get a visa before arrival, especially for people from countries which have a large non-COVID-19-vaccinated population.
What's most likely to happen in the US is that, while the vaccine is never required for all citizens, after a year or two enough people get it that it's virtually a non-issue for the population at large. In the interim, you might see some places "require" a vaccine to enter or travel on that mode of transportation, but it's about as well checked as the current verification processes for "have you been in contact with anyone who's had COVID-19 in the past 14 days, have you had symptoms, etc." - basically, checkmark a box or sign a piece of paper attesting to this, but there'll be no verification of that. Other countries may handle things differently, but the current political structures in the US make it nearly impossible for any sort of universal database to take hold, and so verification will almost certainly be difficult-to-impossible, especially on a large scale.
As for Amtrak? They might "require" it at some point, but it's almost certainly going to be simply added to the text that you checkmark when buying a ticket stating that you'll follow all of Amtrak's rules, or what airlines require when checking in for a flight. Even if Amtrak tries to offer a "vaccinated-people" train car, I don't have a ton of confidence for that to be universally enforced or well-executed. Even then, those sorts of things likely won't happen until we're at a point where everyone who wants a vaccine has been able to get one - until then, it'll be mask wearing and social distancing.