Short story about a mad king who changes his planet's orbit

It's "In the Autumn of the Empire" by Jerry Oltion, Analog October 2009. I can confirm this because I own a copy. ISFDB says that Jerry Oltion is also known as Ryan Hughes and ジェリイ・オルション (Jerii Orushon).

You mention finding it on a site together with "Approaching Perimelasma". That story is by Geoffrey Landis, and first appeared in Asimov's Science Fiction January 1998. It's been anthologised in a lot of places, including Diamonds in the Sky: An Original Anthology of Astronomy Science Fiction edited by Mike Brotherton. "In the Autumn of the Empire" was also anthologised there, but ISFDB doesn't mention any other apppearances.

It's a pleasant surprise that Diamonds in the Sky is available online (also in the Wayback Machine). My link is to its contents page, which shows that all the stories in it are on the site. A comment under the entry for "In the Autumn of the Empire" suggests that it was published in Analog slightly later than in Diamonds in the Sky.

Here are some matches with your description.

A king makes some statement to someone that his planet's seasons are caused by the planet's distance from its star changing:

The emperor of Earth didn't like to be wrong. [...] So when the little common girl, who had been brought to the palace garden to provide a photo op for His Excellence amid the falling leaves, asked him, "Why is there autumn?" two of his attendants faked sudden allergy attacks and ran coughing for the infirmary while another quickly said, "It's because of the tilt of the Earth's—"

Too late. The emperor laughed and said in his reedy voice, "Ah my little darling, that's an easy one. We get autumn because the Earth is moving away from the Sun. Soon we'll be millions of miles away from it, and it'll be winter. [And he then explains spring and summer as getting closer to the Sun.]

When someone points out how wrong he is, he has great quantum engines built to both eradicate the planet's axial tilt and change its orbit into a more eccentric one:

[People who openly criticise his explanation tend to get disappeared, but it is impossible to ignore its logical inconsistencies, such as that it implies winter would happen everywhere on Earth at the same time. And so...] Something had to give, and it wouldn't be the emperor. So nobody was really surprised to find vast engines springing up all over the planet, engines that tapped into the very fabric of space for their power and pushed against that fabric with all their might. [...] Thereafter, the Sun rose directly in the east for everyone on Earth, took exactly twelve hours to cross the sky, and set directly in the west. It did that week after week, with no variation whatsoever. The Earth's axis no longer tilted with respect to the Sun.

Somewhere in the story it is mentioned that an entrepreneur realized this and went into the food storage business, making a fortune as famine gripped the planet:

An astute businessman heard the emperor's pronouncement and immediately bought every cubic foot of refrigerated warehouse space he could, funding it by selling everything he owned in the tourism industry. Then he bought every perishable fruit and vegetable he could lay his hands on, packing them away in his warehouse for a future he hoped would never come.

[And after the vast engines have done their work...] The earth moved on in its orbit, just as the emperor had promised the little girl in his garden. It moved slowly at aphelion, extending winter several weeks longer than usual, but eventually snowbanks thawed the world over. Farmers planted their crops. The growing season was shorter than usual, owing to the Earth's faster orbital speed when nearer the Sun, but there was just enough time for most fruits and vegetables to mature before the weather turned cold again. And the owner of a vast network of refrigerated warehouse space became even wealthier as it dawned on people that an entire planet's worth of perishables would have to be stored at once if they were to avoid a repeat of last winter's famine.

I'm fairly sure that by the end the old king died and the new one made a similarly outlandish claim before, again, resolving to make it come true:

Yes. I won't give it away.