I haven't read the books in a long while, but when I was re-watching the movies recently, I remembered that even though it is depicted as Saruman causing the avalanche/storm on Caradhras, the book hinted that it was the mountain itself not wanting them to pass.
Is there any indication in canon as to why the mountain decided to hamper the way of the fellowship?
In the book, Caradhras is not shown to be sentient. Whether it is merely a natural location where the weather is particularly bad or there is a supernatural force behind this is left unsaid. I think this is deliberate: the world is one where supernatural beings do exist (if trees walk, why not mountains?), and a hostile presence is a conceivable explanation, unlike weather patterns based on fluid mechanics.
In the movie, Saruman is shown to speak to Caradhras, but this does not happen in the book.
Caradhras is introduced in these terms (LOTR II.3):
Yonder stands Barazinbar, the Redhorn, cruel Caradhras;
A few pages later the party comes into sight of the mountain:
On the third morning Caradhras rose before them, a mighty peak, tipped with snow like silver, but with sheer naked sides, dull red as if stained with blood.
There was a black look in the sky, and the sun was wan. The wind had gone now round to the north-east. Gandalf snuffed the air and looked back.
'Winter deepens behind us,' he said quietly to Aragorn. 'The heights away north are whiter than they were; snow is lying far down their shoulders. Tonight we shall be on our way high up towards the Redhorn Gate. We may well be seen by watchers on that narrow path, and waylaid by some evil; but the weather may prove a more deadly enemy than any.'
The setting is ominous (there are other negative signs, such as a flock of black crows). There are possibly monsters living in the mountains, but in Gandalf's words, the deadly enemy is the weather. The opinion of the party is divided:
'We cannot go further tonight,' said Boromir. 'Let those call it the wind who will; there are fell voices on the air; and these stones are aimed at us.'
'I do call it the wind,' said Aragorn. 'But that does not make what you say untrue. There are many evil and unfriendly things in the world that have little love for those that go on two legs, and yet are not in league with Sauron, but have purposes of their own. Some have been in this world longer than he.'
'Caradhras was called the Cruel, and had an ill name,' said Gimli, 'long years ago, when rumour of Sauron had not been heard in these lands.'
And later:
'Caradhras has not forgiven us,' he said. 'He has more snow yet to fling at us, if we go on. The sooner we go back and down the better.'
The party is snowed in. Legolas goes scouting:
'They despaired, until I returned and told them that the drift was little wider than a wall. And on the other side the snow suddenly grows less, while further down it is no more than a white coverlet to cool a hobbit's toes.'
'Ah, it is as I said,' growled Gimli. 'It was no ordinary storm. It is the ill will of Caradhras. He does not love Elves and Dwarves, and that drift was laid to cut off our escape.'
So the place where the Fellowship is is hit particularly hard by snow. This could be due to some malevolent entity controlling the weather, or it could be a natural weather pattern — mountains have sudden, extreme weather events that can be surprising to plains dwellers.
And indeed with that last stroke the malice of the mountain seemed to be expended, as if Caradhras was satisfied that the invaders had been beaten off and would not dare to return. The threat of snow lifted; the clouds began to break and the light grew broader.
"As if" Caradhras was sentient — which isn't to say that he is.
Morgoth raised the Misty Mountains (including Caradhras) specifically as an obstacle for Oromë. Caradhras is charged with the essence of an evil demigod - Sauron's boss and archenemy of the Valar and Maiar (say, Gandalf) - which is ample reason for it to harm and hinder the Fellowship.
Personally, I always took it as an example of the Pathetic Fallacy, an anthropomorphization of nature. Caradhras isn't sentient, but it is a mighty mountain, tall and hard to cross, with nasty weather and dangerous trails. The "cruel" moniker that Gimli gives the mountain seems like the sort of thing that people who frequently suffer from the weather and hardship would say.
With everything else that has been mentioned I thought this deserves to be cited here as well. About the snowstorm on their way up the mountain...
"I wonder if this is a contrivance of the Enemy," said Boromir. "They say in my land that he can govern the storms in the Mountains of Shadow that stand upon the borders of Mordor. He has strange powers and allies."
"His arm has grown long indeed," said Gimli, "if he can draw snow down from the North to trouble us here three hundred leagues away."
"His arm has grown long," said Gandalf.
FotR, The Ring Goes South (pg. 377 for my small paperback edition)
So it's possible that Sauron, not Saruman could have been responsible for the disastrous storm at the foot of the mountain.
@user56's answer is good and I upvoted it. I'd like to add a bit, though, in light of some of the comments and other answers.
I think part of the problem comes from trying to shoehorn LotR (indeed, all of Tolkien's Legendarium) into tidy categories like in D&D: Sentient, Non-sentient, Good, Evil, etc. There are a number of things in Middle-earth which just don't fit into categories or, at least, too little information is provided to judge where they might be placed. Examples: the Barrow-wights, the stone-giants (in The Hobbit), the Watcher in the Water, Old Man Willow, the 'nameless things that gnaw at the roots of the world', Tom Bombadil, Goldberry (and the River-woman, her mother), Shelob, the fox in the Shire woods, and Caradhras.
Some people have explanations for some of them, but they all go pretty far out on a limb of surmise.
Middle-earth is rich enough that multiple explanations are plausible:
People in Middle-earth have anthropomorphized Caradhras — it's just a dangerous mountain
Caradhras is a mountain, but is infused with more than a normal part of Morgoth's malice and is consequently more dangerous than a "natural" mountain.
Caradhras is inhabited by an evil spirit who hates Men and Elves.
Caradhras is a person who hates Men and Elves.
It might be any of them and we'll never know and it's not clear that Tolkien himself ever knew. (Personally, I find the first option to the the most persuasive, but that's all I can say.)
(Compare the comments about Caradhras with Aragon's belief that Anduin, the River of Gondor, would protect Boromir's corpse and not allow it to be defiled.)
C. S. Lewis in Perelandra and That Hideous Strength has Ransom thinking that perhaps the neat division of the world into the Natural and the Supernatural, with the latter hidden far away in the reaches of the distant past and distant future, may be a temporary thing and that the world might change so that they are no longer distinct spheres. In Middle-earth, those spheres do intersect and this makes the world rich and strange.
So, no, there is no indication why or even if.
