We’re not really at a point in our society where blanket enforcement of vaccination by the government is likely to go over well. I have a hard time believing the current administration—or any administration in the foreseeable future—would be daring enough to attempt that. And even if they did, I’m sure it would be tied up in the courts for years.

Why would the government need to order that?

Yeah it seems like there is a straightforward free market solution: you don't get a vaccine, businesses may not allow you on their property or in their stores.

That’s unlikely to happen. Businesses generally have no means of verifying whether you’ve been vaccinated, and any attempt by the government to implement or enforce such a system would inevitably be contested and tied up in courts.

It would be easy to issue certificates to people who have been vaccinated. Businesses already have guidelines for who can enter their premises, like wearing shirts in restaurants in spite of not wearing a shirt being legal.

IANAL, but there’s no way that isn’t going to be contested. That’s not to say it can’t happen, but I have a hard time believing it could happen within a relevant timeframe.

The shirt and shoes rule is a health concern, much like what you’re describing here. Unlike vaccination certificates, though, it’s been around a lot longer and isn’t going to result in a heated political battle anytime soon.