Are renters in the US specifically permitted by law to install airco units that require permanent modification to the property? Or are landlords mandated to install airco units on a renters request? Or..?

The US is a big place, and states are free to set their own rules, and those rules do vary.

I live in Ohio (USA), where a landlord isn't required to provide air conditioning. Some other parts of the US do require it. It's hard to pin down most things in the US using just one generalized statement.

Most of the rentals I've had did provide aircon, in some form, anyway.

For those that did not provide AC, I've just used window air conditioners. They're very annoying, but they work and they aren't usually absurdly expensive. Nobody ever said a word about me using them, nor any the screw holes that I may have left behind by mounting them securely.

I've lived in places that were operated by smaller landlords who I'm quite certain would have been tickled pink to have properly-installed permanent aircon installed -- as long as I asked first and they approved of my plan. But the expectation would have been that it would remain with the rental after I moved out and I'm not necessarily OK with spending that kind of money on a place that I don't intend to occupy for more than a year or three.