Will the government put a law that you can’t go certain places without having the vaccine in you?

The government cannot force you to have a medical procedure.

More likely that businesses and public event arenas will require a proof of vaccine.

And they could legally do that.

That would be much easier than trying to pass an enforce a law on the federal level.

Maybe.  It's already required for kids to be vaccinated to attend public school.

the American government ? No