I believe some places like Walmart and Samsclub will only install H-rated tires for H-rated vehicles.
Many places will not replace your tires with other than the tire specs rated for by the manufacturer. Some will put on a 'better' rated tire. Some will not even allow that. It all comes back to liability and most of the major players (Discount Tire, Costco, etc.) don't want to go there at all.
Although not much of an issue, as a retired accident reconstructionist, I can say that having tires rated at less than the manufacturer recommends can create issues in major accidents. You have to remember that the point where attorneys get involved is the point where, on both sides of the coin, they are trying to push a percentage of 'fault' or 'responsibility'. If the tires on the vehicle have, for example, a lower speed rating than recommended (like a T rating versus a manufacturer recommended H or V) and speed is an issue in the crash, you can bet that it will come up. Handling capabilities are another that I saw come up.
We are not, of course, speaking of fender benders here. These are accidents where there is major injury/damage/death. By definition, an 'accident' is an unintended act that causes death, injury, and/or property damage. The very nature of 'unintended' is the reason why I just stick to the manufacturer recommendations. I sat many a day in court cases where a percentage of fault and or responsibility was assigned for what some would call minor modifications to vehicles. I consider changing a tire spec (many do it to save money, and I certainly understand) when replacing OE tires more than a minor mod as it can open up a 'can-of-worms' that could be avoided.