Yes, they are private entities. I was absolutely floored about 6 weeks ago when one week the Republican mantra was that vaccine passports from the government were an outrageous unconstitutional intrusion into telling private business what they could and could not do, and the very next week Ron Desantis, the Trump heir apparent, signed an executive order preventing businesses from altering, limiting or otherwise taking into account vaccination status. And he was celebrated in GOP circles all over the country for doing so.
THE VERY PRINCIPLE the Republicans claimed as their moral high ground -- the government cannot tell business what to do -- they absolutely trashed the very next week. And then high fived each other. Remarkable.
Schools are a much tougher decision because there are so many complicating factors. For one thing, the students are too young to understand and make informed decisions for themselves as to whether they should be vaccinated and whether it is right to require that others they are around be vaccinated. And parents may have strong feelings for their own child, but you have to balance the risk that decision creates for other children.
I struggle with this one and so am open to all sides. And I say "all sides" because the number of points of view with merit at issue are way more than "both."