Highly doubt that will happen on a large scale. Without legal backing, they just open themselves up to lawsuits. The vast majority of companies have been operating under the assumption that this would happen. With it now settled, it's unlikely they'll risk it.Even if that was possible it would take too long with how drag ass the system works.
Luckily most business aren't that stupid and will most likely just continue their own versions of the mandate behind closed doors or will go back to how they operated before it. My boss sent out email when all this first made it to the SCOTUS to tell us that regardless of the outcome they will "expect" all current and future employees to be up to date on their vaccinations in order to maintain a safe and productive environment. I assume most large businesses (especially those that work in large office spaces with many employees) will do the same.