[Firefoot]

Some quick research has shown a couple places where we can get inside of the characters' heads. Not many, but not none, either.

In the chapter The Uruk-Hai:

'Now,' thought Pippin, 'if only it takes that ugly fellow a little while to get his troop under control, I've got a chance.'

In the chapter Minas Tirith:

Pippin looked at him [Beregond]: tall and proud and noble, as all the men that he had yet seen in that land; and with a glitter in his eye as he thought of the battle. 'Alas! my own hand feels as light as a feather,' he thought, but he said nothing. 'A pawn did Gandalf say? Perhaps; but on the wrong chessboard.'

At the start of the chapter The Houses of Healing:

Slowly the lights of the torches in front of him flickered and went out, and he was walking in a darkness; and he [Merry] thought, 'This is a tunnel leading to a tomb; there we shall stay forever.'

Same chapter:

'I'll never get him there,' thought Pippin. 'Is there no one to help me? I can't leave him here.'
...(Pippin talks to Bergil)...
'I'd better wait here,' thought Pippin.

At the end of the chapter The Black Gate Opens:

Blackness and stench and crushing pain came upon Pippin, and his mind fell away into a great darkness.
'So it ends as I guessed it would,' his thought said, even as it fluttered away; and it laughed a little within him ere it fled, almost gay it seemed to be casting off at last all doubt and care and fear. And then even as it winged away into forgetfulness it heard voices, and they seemed to be crying in some forgotten world far above:
'The Eagles are coming! The Eagles are coming!'
For one mormnt more Pippin's thought hovered. 'Bilbo!' it said. 'But no! That came in his tale, long long ago. This is my tale, and it is ended now. Good-bye!' And his thought fled far away and his eyes saw no more.

Interesting how most of these are Pippin, the exception being one of Merry. Also, most of them come from Book 5. There are places where Sam is talking to himself, at the end of Book 4 and the beginning of Book 6, but I don't know that you could call that getting inside of his head - he is revealing his thought processes, but vocally.