There's nothing that would suggest Bilbo, Frodo, and Sam, but there are other characters who are to some degree self-inserts
Faramir
Tolkien has said that "as far as any character is like me it is Faramir.
I am not Gandalf, being a transcendent Sub-creator in this little world. As far as any character is 'like me' it is Faramir – except that I lack what all my characters possess (let the psychoanalysts note!) Courage.
14 January 1956 draft letter to 'Mr Thompson' (Letters of JRR Tolkien #180)
In particular Faramir is described as seeing a vision of a great wave.
And as they stood so, their hands met and clasped, though they did not know it. And still they waited for they knew not what. Then presently it seemed to them that above the ridges of the distant mountains another vast mountain of darkness rose, towering up like a wave that should engulf the world, and about it lightnings flickered; and then a tremor ran through the earth, and they felt the walls of the City quiver. A sound like a sigh went up from all the lands about them; and their hearts beat suddenly again.
‘It reminds me of Númenor,’ said Faramir, and wondered to hear himself speak.
‘Of Númenor?’ said Éowyn.
‘Yes,’ said Faramir, ‘of the land of Westernesse that foundered, and of the great dark wave climbing over the green lands and above the hills, and coming on, darkness unescapable. I often dream of it.’
The Lord of the Rings - Book VI, Chapter 6 - "The Steward and the King"
This is a recurring dream that Tolkien has had.
For when Faramir speaks of his private vision of the Great Wave, he speaks for me. That vision and dream has been ever with me — and has been inherited (as I only discovered recently) by one of my children.
14 January 1956 draft letter to 'Mr Thompson' (Letters of JRR Tolkien #180)
I say this about the 'heart', for I have what some might call an Atlantis complex. Possibly inherited, though my parents died too young for me to know such things about them, and too young to transfer such things by words. Inherited from me (I suppose) by one only of my children, though I did not know that about my son until recently, and he did not know it about me. I mean the terrible recurrent dream (beginning with memory) of the Great Wave, towering up, and coming in ineluctably over the trees and green fields. (I bequeathed it to Faramir.) I don't think I have had it since I wrote the 'Downfall of Númenor' as the last of the legends of the First and Second Age.
7 June 1955 Letter to W.H. Auden (Letters of JRR Tolkien #163)
Beren
Tolkien modeled the characters of Beren and Luthien on him and his wife. He felt strongly enough about this that he had Beren and Luthien engraved on their tombstone.
I met the Lúthien Tinúviel of my own personal 'romance' with her long dark hair, fair face and starry eyes, and beautiful voice. And in 1934 she was still with me, and her beautiful children. But now she has gone before Beren, leaving him indeed one-handed, but he has no power to move the inexorable Mandos, and there is no Dor Gyrth i chuinar, the Land of the Dead that Live, in this Fallen Kingdom of Arda, where the servants of Morgoth are worshipped.
24 January 1972 Letter to Michael Tolkien (Letters of JRR Tolkien #332)
I have at last got busy about Mummy's grave. .... The inscription I should like is:
EDITH MARY TOLKIEN
1889-1971
Lúthien
brief and jejune, except for Lúthien, which says for me more than a multitude of words: for she was (and knew she was) my Lúthien.
Say what you feel, without reservation, about this addition. I began this under the stress of great emotion & regret – and in any case I am afflicted from time to time (increasingly) with an overwhelming sense of bereavement. I need advice. Yet I hope none of my children will feel that the use of this name is a sentimental fancy. It is at any rate not comparable to the quoting of pet names in obituaries. I never called Edith Lúthien – but she was the source of the story that in time became the chief part of the Silmarillion. It was first conceived in a small woodland glade filled with hemlocks at Roos in Yorkshire (where I was for a brief time in command of an outpost of the Humber Garrison in 1917, and she was able to live with me for a while). In those days her hair was raven, her skin clear, her eyes brighter than you have seen them, and she could sing – and dance. But the story has gone crooked, & I am left, and I cannot plead before the inexorable Mandos.
11 July 1972 Letter to Christopher Tolkien (Letters of JRR Tolkien #340)
John Jethro Rashbold
In the Notion Club Papers, a sort of parody to the Inklings, Tolkien has a character named John Jethro Rashbold. Besides for the similar initials, "Rashbold" is also a literal translation of "Tolkien", and Reuel is one of the alternate biblical names of Jethro.
And then there are some more specific allusions in the text itself where Rashbold appears, as a Pembroke professor of Anglo-Saxon, described as a "grumpy old bear".
In that case, what language is it most likely to be, remembering what Arry told us? Anglo-Saxon. Well, that's not one of my languages, though I know the
elements. So when I'd made a preliminary list of all the separate letters that I could distinguish, I trotted round to old Professor Rashbold at Pembroke, though I didn't know him personally. A grumpy old bear Arry has always called him; but evidently Arry has never given him the right sort of buns.
"The Notion Club Papers" - Night 68 (Sauron Defeated)
It should also be noted that a few other members of the notion club can be read like self-inserts for Tolkien, but not as explicitly intentional as Rashbold.