Do scriptures mention any sort of prayaschitta for people who have chanted the Gayatri Mantra without initiation?
In the language of the Smritis, one who's not initiated into the Gayatri is a Sudra or a Vratya and both are equal as far as rights regarding rituals, chanting Vedic Mantras are concerned.
But, i have read in a book (which is not a scripture but based on them), that if a Sudra (basically take it to mean an uninitiated person and not something derogatory) recites certain Vedic Mantras OMkara, Swaha, Savitri, Vashat etc then he becomes an outcaste immediately. So, this would suggest, that he can remedy that by some easy means.
But, the verses i find in the Smritis are not saying the same. They say the Sudra needs to be punished severely by the king. So, in short, i could not find any expiation measures as such.
By following his own religion even a S'udra enjoys the celestial region. Another's religion should be renounced like the most beautiful wife of another person. (18)
The S'udra, who engages in the recitation [of the Gayatri and the offering of oblations to the Sacred Fire, should be killed by the king ; for by that he becomes the destroyer of the kingdom as the water is of fire. (19)
Recitation [of the Gayatri austerity, and journeys. to sacred places, adoption of a Sannyasa-lite, practice of Mantrams and adoration of Deities, these six bring about the fall of women and S'udras. (135)
Something similar says the Gautama Smriti:
If a Sudra uses abusive language or 18 n.), the part of his body used for the crime should be chopped off. 2 If he has sex with an Arya woman, his penis should be cut off and all his property confiscated; 3 if the woman had a guardian, then, in addition to the above, he shall be executed. 4And if he listens in on a vedic recitation, his ears shall be filled with molten tin or lac; 5if he repeats it, his tongue shall be cut off; 6if he commits it to memory, his body shall be split asunder
Gautama Smriti 11.31.