Did Brigham Young actually teach that the sun and the moon were inhabited?

Brigham Young speculated that the moon and sun were inhabited: he was clearly expressing an opinion

Brigham Young taught that the moon and sun were inhabited,

So it is with regard to the inhabitants of the sun. Do you think it is inhabited? I rather think it is. Do you think there is any life there? No question of it; it was not made in vain.

Brigham is clearly expressing an opinion, and there is no evidence that he is making a prophetic declaration concerning extraterrestrials. He even goes out of his way to indicate that this is what he "rather think[s]," and asks his congregation to consider what they think. He also says that he would want to know if an idea he has is false—even including his religion. These are not the sentiments of a man convinced he must be right by divine gift of prophetic omniscience.

It is particularly ironic that Brigham's remarks were focused on the fact that no one knows much about anything, and so humility is appropriate on most questions. Critics have taken this wise stance, and have tried to invert Brigham's intent—changing him from an advocate of humility before the unknown into a doctrinaire know-nothing who is certain of absurdities. The critics might do well do follow Brigham's example.

Brigham Young made the following statement in 1869:^[1]

It has been observed here this morning that we are called fanatics. Bless me! That is nothing. Who has not been called a fanatic who has discovered anything new in philosophy or science? We have all read of Galileo the astronomer who, contrary to the system of astronomy that had been received for ages before his day, taught that the sun, and not the earth, was the centre of our planetary system? For this the learned astronomer was called "fanatic," and subjected to persecution and imprisonment of the most rigorous character. So it has been with others who have discovered and explained new truths in science and philosophy which have been in opposition to long-established theories; and the opposition they have encountered has endured until the truth of their discoveries has been demonstrated by time...

I will tell you who the real fanatics are: they are they who adopt false principles and ideas as facts, and try to establish a superstructure upon, a false foundation. They are the fanatics; and however ardent and zealous they may be, they may reason or argue on false premises till doomsday, and the result will be false. If our religion is of this character we want to know it; we would like to find a philosopher who can prove it to us.

The context for Brigham's remarks, then, are that new ideas and truths are often mocked or rejected by those who cling to older ideas. And, were he to have such an idea, he would want to know.

He then says:

We are called ignorant; so we are: but what of it? Are not all ignorant? I rather think so. Who can tell us of the inhabitants of this little planet that shines of an evening, called the moon? When we view its face we may see what is termed "the man in the moon," and what some philosophers declare are the shadows of mountains. But these sayings are very vague, and amount to nothing; and when you inquire about the inhabitants of that sphere you find that the most learned are as ignorant in regard to them as the most ignorant of their fellows.

Brigham goes on to speak about inhabitants of the moon. In context, his point is clearly that no one;—even experts—knows very much about the universe. There are many things (such as whether the moon is inhabited) about which no one of his day could speak clearly.

It then becomes very clear that Brigham is expressing his personal views, not laying down divine truth from on high

So it is with regard to the inhabitants of the sun. Do you think it is inhabited? I rather think it is. Do you think there is any life there? No question of it; it was not made in vain.

Brigham is obviously expressing his opinion, but his point remains that no one knows very much about such things. To reject a novel idea simply because it is new—such as Mormonism—is irrational. All true ideas were once new, and treated with suspicion.

William Herschel—the preeminent astronomer of his generation and the man to discover Uranus—was also firmly of the belief that the sun was inhabited.^[2] One author wrote:

Herschel was not a raving amateur. A gifted astronomer, he discovered Uranus, and was the first to realize that sunlight included infrared light as well as visible light. His sister, Caroline, became famous in her own right for discovering comets, so he did not lack for intelligent conversation. He just had his own theories. Herschel believed that life existed on every celestial body in the universe. He was aware that the sun people saw was too hot to support life. He just assumed there was something underneath that burning atmosphere. When he observed sunspots, he believed that they were openings in the atmosphere, or perhaps mountains, and that if people could get a close look at the planet beneath, they would be able to spot signs of life. Herschel was not alone in his beliefs - as more information on the sun turned up, astronomers speculated on how it would affect life on the surface of the sun, and what kind of life might survive in those environments.^[3]

Church publications did not shy away from embracing later scientific findings on the matter

Church publications did not shy away from embracing later scientific findings on the matter:

1856

Desert News noted:

Proof that the Moon is not Inhabited.

"Dr. Scoresby, in an account that he has given of some recent observations made with the Earl of Rosse’s telescope, says: ‘With respect to the moon, every object on its surface of 100 feet was distinctly to be seen; and he had no doubt that, under very favorable circumstances, it would be so with objects 60 feet in height…. But no vestiges of architecture remain to show that the moon, is, or ever was, inhabited by a race of mortals similar to ourselves….. There was no water visible…."^[4]

1880

"As there is no air nor water on the moon, but very few changes can take place upon its surface. There can be no vegetation and no animals, and although many astronomers have brought their imaginations to bear upon this subject, and have given us descriptions of the beautiful scenery upon its surface, and have even peopled it with inhabitants, we have every reason to believe that it is as barren and lifeless as an arid rock."^[5]

Joseph Smith and Moon Quakers?

Summary: A late third-hand account attributes a similar idea to Joseph Smith.