In Arabic, the term infallibility, is called Asma (عصمة ,(of which the adjective is Ma’soum (معصوم ,(i.e. infallible. We do not find this word in the Quran with the same meaning, but it means protection from sin, from temptation:

O Messenger. proclaim the (message) which hath been sent to thee from thy Lord. If thou didst not, thou wouldst not have fulfilled and proclaimed His mission. And Allah will defend thee from men (who mean mischief). For Allah guideth not those who reject Faith. (Quran 5:67)

In Sunni schools, mortal man is considered fallible, so normally no one can be infallible except the one chosen by Allah. They are the apostles and prophets whose task is to convey the Message. In their case, we can distinguish two stages of life, which are the one before the inspiration and what follows the inspiration. Before inspiration they are just as mortal humans as anyone else, so they are fallible and even commit sins. Think of the case of Moses, who killed a man and had to flee. Or the story of Jonah. Then comes the inspiration and from then the life of the prophet will change radically.
Otherwise how could a person be followed who conveys the word of Allah to people if he could make mistakes, commit faults? If this were the case, he would be no example for humanity:

Ye have indeed in the Messenger of Allah a beautiful pattern (of conduct) for any one whose hope is in Allah and the Final Day, and who engages much in the Praise of Allah. (Quran 33:21)

… So, take what the Messenger assigns to you, and deny yourselves that which he withholds from you. And fear Allah. for Allah is strict in Punishment. (Quran 59:7)

Obey Allah, and obey the Messenger, and beware (of evil): if ye do turn back, know ye that it is Our Messenger's duty to proclaim (the message) in the clearest manner. (Quran 5:92)

So far, all Sunni and Shiite schools share similar principles. From then on, however, the two perceptions differ. The Sunni schools admit the infallibility of the Prophets, whose line is closed by Mohammed (peace be upon him). In the Shiite conception, however, this line is followed by the household of the Prophet (peace be upon him) and by the imams, on whom I quote the following reference:

“By definition, “infallibility” in Islam does not mean incapability of committing sin or erring such as in the realm of angels. The Infallible means he who by the will of God abstains absolutely from all that is forbidden. Therefore “infallibility” in the realm of humankind means the infallible person enjoys a divine inspiration, by which he voluntarily yet absolutely protects himself against any sin or error. Thus, practically, it is impossible for a Ma’soum to commit a sin or make a mistake.” https://www.al-islam.org/articles/infallibility-mansour-leghaei

The interpretation of infallibility also led to differences in the structure of Sunni and Shiite schools of jurisprudence. The Sunni schools consider the Quran and the Sunna as the basis and from there are derived the different opinions which are reconciled and as a result the Ijma’, the majority of scholars, i.e. the Shura, the council takes the decision, as far as no one is infallible. In the reconciled opinion can occur the least chance of error. In the Shiite conception also the Quran and the Sunnah are the basis, although there is a difference in regard to Hadiths. However, it is not the Ijma’ but the Marja', the scholar who takes decision. He is the source of knowledge according to this perception. Based on the above definition taken from the Shiite reference, Marja' is the Islamic Authority, i.e. HujatulIslam, who is the most authentic in terms of infallibility, since the constant pursuit of faith, his inspired existence, his knowledge protects him from committing sins. The Mardja’s often establish schools.

And remember that Abraham was tried by his Lord with certain commands, which he fulfilled: He said: "I will make thee an Imam to the Nations." He pleaded: "And also (Imams) from my offspring!"
He answered: "But My Promise is not within the reach of evil-doers. " (Quran 2:124)

As Quran is saying that the Imams are appointed by God, but according to the Shiite perception the Asma is also revealed to the Imams.