Which Purana talks about Sahasrakavacha (one who has a thousand armors) - the purva janmam (past life) of Karna?
The Padma Puran talks about Sahashrakavacham, the previous birth of Karna.
During a fight between Shiva and Brahma, Shiva chopped off Brahma's head and some drops of sweat appeared on the forehead of Brahma out of anger. When he wiped out those drops of sweat out of it was born Sahasrakavacha (the person with thousand armors):
svedataḥ kunḍalī jajñe sa dhanuṣkomaheṣudhiḥ
sahasrakavaca viraḥ kimkaromityuvācaha [Pdm. Pu. - 1.14.4]
From the sweat wiped by Brahma was born a person with bow and thousand armors. Then he asked, "What shall I do?".
The story is a bit long, but in short Brahma told Sahashrakavaca to fight with Shiva. So Shiva went to Vishnu for asking help. Then in some course of events Vishnu produced another warrior with thousand hands from a skull hold by Shiva by filling it with His blood. He named this person as Nara and he started fighting with Sahasrakavacha. The fighting continued until only last kavacha of Karna and two hands of Nara remained. Then Vishnu went to Brahma and told who will win. Brahma told Nara will win in the next birth as Arjuna.
Then the fighting stopped as both were told to be given the chance for fight in the next birth. Vishnu then told Sun to cause the birth of Karna through Kunti and Indra to cause birth of Arjuna. So this is the story of previous birth of Karna in short. I'll add more when get time.
TL;DR There is no scriptural evidence for Sahasrakavacha being the previous birth of Karna.
As far as I'm aware, the only place in Hindu scripture that discusses Dambodhbhava, AKA Sahasrakavacha, is the Udyoga Parva of the Mahabharata. As I discuss in this answer, when Krishna went as an envoy of the Pandavas to the Kauravas' palace, the Kauravas remained silent after Krishna made his offer of peace. So Vishnu's other incarnation Parashurama, who happened to be in palace at the time, admonished them by recounting the story of Dambhodbhava:
There was a king of yore named Dambhodbhava, who was the Head of the earth. It hath been heard by us that his sovereignty extended over the whole world. And that mighty car-warrior, rising every morning after the night had passed away, called the Brahmanas and the Kshatriyas unto himself and asked them, saying, "Be he a Sudra, a Vaisya, a Kshatriya, or a Brahmana, is there any one who is superior or even equal to me in battle?" ... And some high-souled Brahmanas then ... told him, "There are two persons who are foremost of all men and who are always victorious in battle. Thou, O king, wilt by no means be equal to them if thou seekest an encounter with any one of them." ... And the two Rishis received the king hospitably, with fruits and roots, and a seat and water.... [T]he king said unto them the same words that he was in the habit of saying unto all. And he said, "The whole earth has been conquered by the might of my arms. All my foes have been slain. Desiring a battle with you both I have come to this mountain. Offer me this hospitality. I have been cherishing this wish from a long time." Thus addressed, Nara and Narayana said, "O best of kings, wrath and covetousness have no place in this retreat. How can a battle, therefore, be possible here? There are no weapons here, and nothing of unrighteousness and malice. Seek battle elsewhere. There are many Kshatriyas on earth."
Although thus addressed, the king still pressed them for giving him battle. The Rishis, however, continually soothed him and overlooked his importunity. King Dambhodbhava, still desirous of battle, repeatedly summoned those Rishis to fight. Nara, then, O Bharata, taking up a handful of grass-blades, said, "Desirous of battle as thou art, come, O Kshatriya, and fight! Take up all thy arms, and array thy troops. I will curb thy eagerness for battle hereafter!" ... Dambhodbhava with all his troops, desirous of slaying that ascetic, covered all sides with a shower of arrows. That ascetic, however, by means of those blades of grass, baffled all those terrible shafts of the king that were capable of mangling the bodies of hostile warriors. The invincible Rishi then let off towards the king his own terrible weapon made of grass-blades and which was incapable of being counteracted. And highly wonderful was that which happened, for that ascetic, incapable of missing his aim, pierced and cut off, by those grass-blades alone, the eyes and ears and noses of the hostile warriors, aided also by his power of illusion.
And beholding the entire welkin whitened by those grass-blades, the king fell at the feet of the Rishi and said, 'Let me be blessed!" Ever inclined to grant protection unto those that sought it, Nara then, O king, said unto that monarch, "Be obedient to the Brahmanas and be virtuous. Never do so again.... Blessed be thou, and with our leave, go hence, and never again behave in this way. At our command, enquire thou always of the Brahmanas as to what is for thy good!" The king then, worshipping the feet of those two illustrious Rishis, returned to his city, and from that time began to practise righteousness. Great indeed, was that feat achieved of old by Nara. Narayana, again, became superior to Nara in consequence of many more qualities. They that were Nara and Narayana in days of yore are now Arjuna and Kesava.
(Reminds you a bit of Snow White, doesn't it?) As you can see, there's no mention of Dambhodhbhava reincarnating as anyone, only about Nara and Narayana reincarnating as Arjuna and Krishna. And there's not even a mention of him having 1000 coats of armor, which is the whole reason people call him Sahasrakavacha.
One work that does mention the thousand coats of armor is the Narayaneeyam, a medieval poem summarizing the Srimad Bhagavatam; here is what it says:
The Asura known as Sahasrakavacha, had one thousand coats of armour. One who did penance for a thousand years and also fought with the Asura for a thousand years, simultaniously, only could pierce the armours. Thou as Naaraayan along with Nara did the required in turns and all the coats of the Asura were destroyed but one remained. Then Thou killed him effortlessly, as a sport.
But again, here too there's no mention of Karna. As far as I can tell, the earliest suggestion that he was reborn as Karna is in the Tatpara Nirnaya, Madhvacharya's commentary on the Mahabharata. Madhvacharya says this:
Because of the stigma attached to him in his becoming instrumental to eliminate his own brother vaali, sugriiva took birth as karNa, together with the lifeforce of a demon named sahasra-varman.
But I'm not sure where Madhvacharya got this story from.