Lord Jagannatha is Vishnu himself. This is confirmed both by puranic and historical evidence.
Brahma Purana describe Jagannatha
After taking the holy dip in the five Tīrthas, Purusottama should be visited. Even that person who is far off but who devoutly glorifies Purusottama everyday with a pure mind shall go to the city of Visnu. The man who undertakes pilgrimage with faith and concentration to Krsna (shrine) shall be liberated from all sins. He shall go to the world of Visnu. By visiting the wheel of Hari, that is kept above the palace even from a distance and by bowing to it devoutly, the man is immediately liberated from sins.
- Chapter 48, Brahma Purana.
Narada Purana describe Jagannatha
The second idol resembled the blue cloud in complexion. It had eyes resembling lotuses. The lord resembled the flower of Atasi (Hemp or flax) and his eyes were as large as the petals of the lotus. He was Hari, Acyuta, with the Srivatsa mark on the chest, with shining yellow robes. He was divine in character, holding the discus and the conch. He dispelled all sins.
- Chapter 54, part 5, Narada Purana.
Skanda Purana describe Jagannatha
All those Devas who had come for the sake of the Yajnas were free from ailments, O Brahmanas. On seeing Hari becoming visible they worshipped him. Just as they worshipped Madhava before, so also they worshipped the tree of Visnu. In the course of the worship of Devas divine symbols appeared. On the fifteenth day the Lord completed the work. In the manner explained by me before the Lord manifested himself in four forms. It has been described to you before. He was seated in the divine throne along with Bala, Subhadra and Sudarsana. Janardana has conch, discus, iron club and lotus shining in his hands.
- Chapter 19, Purushottama Ksetra Mahatmya, Skanda Purana.
Padma Purana says
He who stays at the Nila (mountain) is Rama, Visnu (himself). Sita is actually the great Laksmi, the cause of causes. He whose name is prescribed in (i.e. as) the expiation for the murder of a brahmana, will, after performing the horse-sacrifice, purify the worlds.
- Chapter 22, Patala khanda, Padma Purana.
Nila mountain is Nilachala where Jagannath temple is located. As Padma Purana says, the Lord who resides in Nilachala is Hari himself.
One of the oldest inscriptions about the temple calls it the tirtha of Hari
Another inscription refer to the temple as the home of Krishna
Another inscription from the time period of Anantavarma Chodaganga again refers to the God as Krishna
Another inscription calls the deities of Puri as Hali, Chakri and Subhadra
Hali is Halayuddha or Balarama. Chakri is clearly Krishna/Vishnu and Subhadra is Subhadra. Interestingly this inscription is from king Anangabhima deva who despite being a Vaishnava also adopted Shaiva and Shakta titles of Rudraputra and Durgaputra. Still even he identifies Jagannatha as Lord Vishnu alone and none else.
Thus there should be no doubt that Lord Jagannatha is Hari himself.