Part of my story will be set on a planet where a culture similar to that of the Old West has arisen - cowboys, frontier towns, homesteaders, etc. In order to make this development more plausible, I want the entire surface of the planet, or large swathes of it, to be fairly similar in terms of climate and general nature to the Great Plains of America. The most important aspects here are the plain landscape, the infertile, dry soil, the extremely hot summers and freezing winters, the violent storms. Is there any specific scenario - a given orbit, a given star type, a given planet type, some set of planetary or solar characteristics - that would make this plausible?
In lieu of any other answers being written, the answer to your planetary configuration is primarily a matter of water.
Some basics. The reason why living on the coast is so temperate (warmer nights, cooler days) to living in deserts and inland environments is that it is close to a massive body of water called the Ocean. Water is an ideal thermal mass; that is to say, water absorbs heat from the environment when it's hotter than the water, and then releases it when the environment cools. Deserts have the reputation of being really hot, but at nights they are also really cold because there is nothing slowly releasing the heat through the night like water does on the coast.
The thing is, if you have a desert planet, you don't have enough water to support life. So, what you need is a planet on which the water does not evaporate as quickly. This means you either need a planet like Earth in terms of climate but with little water (say around 20% of the surface covered in a large central ocean) or a planet a little further out from the sun so it's colder, but carries more water to trap what heat comes in but without massive evaporation / condensation cycles that we have here on Earth.
As has been mentioned in comments, you would want less water than Earth still, but you could get away with significantly more ocean as a percentage the further out you are from the sun, so long as your planet is still in the goldilocks zone.
A final thought is that if your planet has a high concentration of greenhouse gases in its atmosphere, especially CO2, you're probably able to go out even a little further still, but the history of the earth tends to indicate that you shouldn't rely on such a mixture as a permanent feature of your atmosphere as it can change significantly over geological epochs. Better to be closer in where atmospheric gas ratio fluctuations can't cause global extinction events by freezing a planet over completely.
In short though, the rule of thumb is that the less water (thermal mass) you have in your environment, the bigger the difference you'll see between summer and winter, day and night.