What is the Biblical and Traditional basis for doing Penance for others?

In Catholicism such people who take on penances for others are called victim souls. Victim Souls are someone who offers their life, their will and their intense sufferings to God for the salvation of immortal souls.

To speak of certain holy individuals whom we collectively call "victim souls" takes us into territory that many Christians often fail to understand: the redemptive power of human suffering.

A victim soul is an individual who has been chosen by God to undergo physical, and sometimes spiritual, suffering beyond that of normal human experience. The victim soul willingly accepts this unique and difficult mission of offering up his or her pains for the salvation of others.

Although Jesus Christ accomplished our redemption once and for all by suffering torture and crucifixion for our sins, Scripture also affirms the value of human suffering. Christ points to this value himself when He says, "Whoever wishes to come after me must deny himself, take up his cross, and follow me" (Mk 8:34).

Perhaps more pointedly, St. Paul writes that we are "heirs of God and joint heirs with Christ, if only we suffer with him so that we may also be glorified with him" (Rom 8:17).

St. Paul's own example gives clear witness to the possibility of "offering up" our suffering for the benefit of others when he declares, "Now I rejoice in my sufferings for your sake, and in my flesh I am filling up what is lacking in the afflictions of Christ on behalf of his body, which is the church" (Col 1:24).

Our own suffering takes on a redemptive dimension when we unite it with the passion of Christ.

"To be a victim necessarily implies immolation, and as a rule, atonement for another.

"Although strictly speaking one can offer oneself as a victim to give God joy and glory by voluntary sacrifice, yet for the most part God leads souls by that path only when He intends them to act as mediators: they have to suffer and expiate for those for whom their immolation will be profitable; either by drawing down graces of forgiveness on them, or by acting as a cloak to cover their sins in the face of divine justice.

"It stands to reason that no one will on his own initiative take such a role on himself. Divine consent is required before a soul dares to intervene between God and His creature. There would be no value in such an offering if God refused to hear the prayer." - Sister Josefa (What is a Victim Soul?)

Victim souls are co-redeemers with Christ and there are several individuals in the history of the Church that are considered Victim Souls.

St Faustina Kowalska, St (Padre) Pio of Pietrelcina, Therese Neumann, Blessed Alexandrina Maria da Costa, Marie Rose Ferron, Marthe Robin and St Gemma Galgani are a just a few examples of Victim Souls.

A “victim soul” is someone who offers their life, their will and their intense sufferings to God for the salvation of immortal souls. By uniting their sufferings to Jesus’ sufferings, they bring down torrents of graces upon souls. Bl. Alexandrina was told by Our Lord: “Thousands have been saved by your terrible sufferings.” Many victim souls received revelations about God’s love and mercy. Here is one touching example: “I want to make known in writing the intense love with which My Heart burns for souls; I want to complain that I am forgotten, rebuffed; I want to plead for love as a beggar pleads for a crust of bread; I love souls so much, yet very often I am not understood and not loved.” – Jesus to Bl. Dina

We are not all called to be “victim souls”. St. Padre Pio said to one of his spiritual daughters that relatively few are called to be victim souls. - What We Can Learn From Victim Souls

The Vicitim Soul?

The Catholic Church does not officially designate anyone as a victim soul. The term stems, rather, from the testimony of those who have encountered Christians who seem to undergo the kind of redemptive suffering we have described.

The victim-soul status, even when it is genuine, is a matter of private revelation. Consequently, the Church teaches us that we are not obliged to accept, as part of the Catholic faith, the legitimacy of any particular person for whom such a claim is made, nor the genuineness of any mystical or miraculous claims that have been made in connection with such a person.

This includes even those people whose private revelations have been approved by the Church. (See the Catechism of the Catholic Church on this subject, No. 67).

Nevertheless, the fundamental spiritual truth -- illustrated so vividly by the notion of a victim soul -- still remains: We can accept the pain and suffering of this life with patience and love for the intentions and benefit of ourselves and others in the Communion of Saints. - Are We Obliged to Believe?

Outside this particular aspect of there are some of the faithful who performed penances for others as a sort of "proxy penances". An example of this can be found in the life of St. John Vianney (Cure d'Ars). It is described in the Biography of St. Jean-Marie Vianney by Francis Trochu. We all know the penances of this man of God performed on himself!

His diet was a self-inflicted penance: boiled potatoes, usually days old, never more than two in a day. If someone gave him a loaf of bread, he would exchange it for a crust from a beggar. To his penitents, on the other hand, he gave easy penances, saying "I give them a light penance and perform the rest myself." - St. John Marie Vianney, Patron of Parish Priests

Doing proxy penances are not something new in the history of the Church. It is known that some penitents have asked of to do hard penances, like making a long pilgrim in one's place because of ill health in the Middle Ages.

This article investigates the religious practice of suffering for others in the early Middle Ages. In proxy penance, one person completed a penitential work for another, who received the spiritual benefit. This practice was based on the idea that one person could stand in for another to bear his burden. Using penitential, conciliar, liturgical, and epistolary sources, I uncover two types of proxy penance. First, priests shared in the penance of those who confessed to them. Liturgical texts include Masses in which the priest completes the penance for someone who could not complete it himself. Penitential texts admonish the priest to “share in the foulness” with the sinner in order to bring about the remission of his sin. Second, there was both a promotion and a criticism of proxy fasting among the laity. This sic et non rhythm shows that early medieval penitential culture could not control the demand for proxy penance. Some attention is also paid to the practice of proxy penance in the eleventh-century monastic milieu of Peter Damian. This article broadens the scope of current scholarship on penance by focusing on its substitutionary ability. Also, this article explores the changing notions of and metaphors about sin in this period—from medical to economic—that fueled proxy activity. - Penitents and Their Proxies: Penance for Others in Early Medieval Europe