Free will permeates Tolkien's cosmos and drives everything that happens in LotR. For Tolkien, free will defines us as humans. Catholic belief would support this, as free will is considered part and parcel of our being made in the image of God. Free will is so important that God allows us to sin rather than take it away from us, an idea Tolkien uses in his story of creation and fall in The Silmarillion. Tolkien sees anything that has the aim of depriving another person of the use of his or her free will as - by definition - evil, a concept that plays a fundamental role in LotR.
Some authors' characters tend to be more controlled by outside forces, while Tolkien's are given more freedom to choose their own actions. All Frodo would have had to do to escape the journey to Mount Doom is sit quietly at the Council of Elrond. I'd attribute this difference in large part to Tolkien's emphasis on free will; he wants it to be clear that when a call is answered, it's answered freely. (In an essay elsewhere, I contrasted this with Orson Scott Card's Ender's Game, in which the main character is controlled and lied to throughout the book and is ultimately tricked into performing actions he very well might have refused if given a truly free choice.)
Among the main characters in LotR, I can't think of one who doesn't have some freedom to choose his or her own path. That doesn't mean the character necessarily knows what's at the end of the Road, but each has the ability to accept the risk or turn it down. Théoden doesn't know that he'll die on the Pelennor Fields, but he freely chooses to accept that as a possibility when he decides to go to war himself instead of just sending others. Aragorn's statement about Merry in "The Passing of the Grey Company" could be applied to any of the heroes in LotR: 'He knows not to what end he rides; yet if he knew, he still would go on.'
It's because of all those free choices that there are so many "what if's" in LotR. There are myriads of ways the story could have played out if one or another character had at any point made a different choice than the one that's in the book. What if Faramir, instead of Boromir, had left Gondor to search for Imladris? What if Éowyn had stayed at Meduseld "doing her duty"? What if Frodo had sat quietly at the Council of Elrond, or even refused to give Gandalf the answer he "hardly expected" back at Bag End? There are countless possibilities based on the free-will choices of the characters, from the heroes to those barely mentioned (What if Ghân-buri-Ghân had told his people to ambush the Rohirrim instead of making a pact with them?).
There's an event in LotR that draws Tolkien's philosophy about free will into one tight, unbearable point. In "The Breaking of the Fellowship," Frodo writhes in pain between the Eye (Sauron) and the Voice (Gandalf):
The two powers strove in him. For a moment, perfectly balanced between their piercing points, he writhed, tormented. Suddenly he was aware of himself again. Frodo, neither the Voice nor the Eye: free to choose, and with one remaining instant in which to do so. He took the Ring off his finger.
When Frodo becomes "aware of himself" he also becomes aware of his freedom to choose; his free will is an integral part of "himself."
When he died, Tolkien still hadn't found a way to fit orcs into this framework - at least not one that completely satisfied him. Orcs are obviously "people," in that they think and make at least some rudimentary decisions for themselves. But do they have free will? If so, then it is up to each orc to make the choice to follow good or evil. Judging from the orcs we meet in The Hobbit and LotR, this is difficult to imagine. Tolkien did believe it was possible for even good people to be broken to the point where they no longer have that fundamental choice. He compared Frodo's situation at the Sammath Naur not only to someone being crushed by a boulder but also to prisoners who lose the use of their free will through being brainwashed. He says that Frodo's was not a 'moral failure'; neither would be evil acts committed by someone because he or she has been deprived of free will. The evil - and Tolkien considered it to be a great evil - would be committed by whoever was responsible for that loss of free will.
If Tolkien himself couldn't find a completely satisfying answer to the question of orcs' free will, I'm certainly not going to claim that I have, but there are some interesting clues when Sauron's power and will is taken from them. As Frodo claims the Ring at Orodruin, the following happens at the Black Gate:
...even at that moment all the hosts of Mordor trembled, doubt clutched their hearts, their laughter failed, their hands shook and their limbs were loosed. The Power that drove them on and filled them with hate and fury was wavering, its will was removed from them; and now looking into the eyes of their enemies they saw a deadly light and were afraid.
Perhaps even more telling is this, just a few paragraphs later, when the Ring is destroyed:
As when death smites the swollen brooding thing that inhabits their crawling hill and holds them all in sway, ants will wander witless and purposeless and then feebly die, so the creatures of Sauron, orc or troll or beast spell-enslaved, ran hither and thither mindless...
But - and a very important "but" it is:
But the Men of Rhûn and of Harad, Easterling and Southron, saw the ruin of their war and the great majesty and glory of the Captains of the West. And those that were deepest and longest in evil servitude, hating the West, and yet were men proud and bold, in their turn now gathered themselves for a last stand of desperate battle. But the most part fled eastward as they could; and some cast their weapons down and sued for mercy.
The reactions of "creatures of Sauron" and those of Men seem to be fundamentally different. No matter which response individual men have to the dissolution of Sauron, they do not become mindless when his will is withdrawn from them. He may have enslaved and lied to them and bent their wills to his (so to what degree they're responsible for their choices is something only The One would be able to judge), but they are still children of Ilúvatar and not creatures of Sauron. The most important thing about that "But" is that Tolkien saw fit to insert it. I'm not sure every author would have sensed the need to show this fundamental difference between the two groups of creatures.
(The story of the creation of the Dwarves is also relevant here, it seems to me, because it says quite clearly that any creatures that are created by anyone but Eru are devoid of free will because their maker has only his own will to give them. Because their creator repents, the Dwarves have a happier outcome than the orcs.)
Free will is about freedom, and freedom involves trust. Sauron isn't the only "bad guy" who's kept his underlings from knowing enough about his plans to be able to thwart them. There's "no honor among thieves" because the thieves can't trust each other to behave honorably. The problem with having mindless underlings is shown in the responses at the Black Gate - after the two hobbits whom the Wise have trusted with the fate of Middle-earth reach the end of their journey.
Tolkien also trusted his readers, so he defended their right to the freedom of applicability - as opposed to the "purposed domination of the author" evident in allegory. He trusted the story itself as it grew and became something other than he had planned. Can it in some way be said that he must have had trust in his characters - trusted, that is, that the decisions they made would show him "the way it really happened," eliminating the need to control what they did by trapping them into behaving the way he wanted them to? In early drafts of LotR, Frodo (or Bingo, as he was called then) begins his journey as a lark, not a mission. He doesn't understand the danger until he's in the midst of it - until it's too late to avoid it. Somewhere during the writing, it seems Frodo let the author know that he was mature, aware, loving and courageous enough to make a free decision to leave the Shire, and, ultimately, to take on the further Quest for which he'd been gifted and graced. He didn't need to be forced, trapped, or tricked into it. (As someone who's written some fiction, I can tell you that characters do have the ability to tell you when something's not quite right in their story.)
In his "Postscript" to his book Tolkien and the Great War, John Garth compares and contrasts Tolkien's post-war writing with some of the more common types of literature that came out of World War I. He notes that many of the veterans of that war were unhappy with some of the memoirs and poetry that dealt with its horrors, because they felt it portrayed the soldiers as passive victims, which wasn't how they thought of themselves. Garth contrasts this with Tolkien's approach, in which even those forced into horrible circumstances retain the ability to make decisions about how to face them. Garth doesn't examine what he thinks are the reasons behind this difference, but Tolkien's approach is compatible with what his religious faith tells him about what it means to be human. As evidenced in his letters, Tolkien was very aware that even the best people can be broken to the point where they no longer exercise free will, but he doesn't emphasize that in his stories or show it to be the norm. The one obvious case of it is Frodo claming the Ring at the Sammath Naur, and Tolkien's letters that talk about the subject are in response to readers' reactions to that specific event. In fact, the importance Tolkien places on free will makes it even clearer that Frodo's complete inability to exercise it is an extreme situation rather than the way life usually works. It shows that he's given everything to the Quest - including his very personhood.