According to the Manusmriti, a Vratya is one who has not undergone upanayana (initiation) within the prescribed time limits of respective varnas and hence a Vratya becomes an outcaste. This is mentioned in Manusmriti 2.39:
After those (periods men of) these three (castes) who have not received the sacrament at the proper time, become Vratyas (outcasts), excluded from the Savitri (initiation) and despised by the Aryans
However, what if the son of a Vratya wants to undergo upanayana. Is he allowed? If a prayaschitta (atonement) needs to be undergone by the son of the Vratya, please mention what is the prescribed prayaschitta and how to perform it.