I'm assuming we're working with the Americas here? There are very few of them, and they come with an asterisk for what we mean by 'native.' I'll focus on the accounts for the conquests of Mexico and Peru, but hopefully other people have information on other areas.

As you may be aware, none of the existing American societies had writing. There were some like the Maya with pictographs, but they weren't equivalent to the sort of writing Europeans came with. (1) The native accounts of encounters with Europeans and of their pre-colonial history were predominantly oral histories. As you can imagine when 90+% of a native population is wiped out by disease or war, that oral record was absolutely decimated.

That being said, we do have some things. Most of them come from Spanish scholars in the years after the conquest or from mestizo and native writers in that period.

Sahagun's Florentine Codex is the big one for Mexico. (2) It's a compilation of mostly orally-sourced accounts from the natives after the conquest. The obvious qualification for this one is that it's a Spanish writer putting it to paper after the conquest, colouring it with his own perceptions and ideas of what history is. However, it does make a concerted effort to recount a Mexican view of the conquest that contrasts with that provided by Cortes or Bernal Diaz.

It's a good idea to mention Broken Spears, a compilation effort by Miguel Leon-Portilla that pulled together extracts from the Codex and the patchy other sources to create a more readable account of the Spanish conquest and the events leading up to it from the native perspective. I'd recommend it for you because you're interested in this topic, but to be honest I'd recommend it in general. It's easily one of the most vivid and interesting histories I've ever read; it often reads more like poetry than history and is absolutely worth your time. (3)

Peru and the Inca have Juan de Betanzos. He was a conquistador that ended up marrying one of he ex-wives of the Inca ruler Atahualpa, called Dona Angelina. He used her and other surviving Inca elites to write a remarkably comprehensive history of the Inca based on these sources. He covers events from the semi-mythical founding of Cusco, through the likes of Pachacuti, a detailed account of the Civil War on the eve of the conquest, and then the events of the conquest itself. Again, it's a great read that I'd highly recommend. (4)

Another one worth noting is Garcilaso de la Vega. He was a mestizo writer born to a Spanish father and an Inca mother who wrote histories of the Inca and of the Spanish expedition into Florida. It's quite pro-native in its tone, especially the account of Florida, but I've only got a second-hand familiarity with it myself. (5)

Unfortunately, there's not much else. There weren't natives writing during the conquest. While we do have things like the work of Betanzos and Sahagun, they do face a critical issue of being native accounts filtered through a Spanish lens. What content they include or omit, how they frame it, how they translated it from Quechua or Nahua into Spanish - all of it raises a question of just how authentic to the native experience it really was.

They definitely are distinct from the dominant Spanish narratives though. The conquistador's belief that all the natives saw them as Gods doesn't get the same credence from the native sources. (6) Figures like the translator La Malinche that fade into the background of the Spanish sources are instead quite prominent in the native ones. There's also some clear distaste towards the Spanish that does suggest these sources do preserve some native sentiment free of Spanish censorship. There's this lovely snippet from Broken Spears that captures this:

"The truth is that they longed and lusted for gold. Their bodies swelled with greed and their hunger was ravenous; they hungered like pigs for that gold.” (7)

So there are some sources that capture something of the native perspective. There's obviously much more from the European side, but we're not completely lacking for Mexico and Peru at least.

Footnotes

(1) J.H. Rowe, ‘Inca Culture at the time of the Spanish Conquest’, in Handbook of South American Indians, vol. ii, (1956), p201

(2) Fray Bernandino de Sahagun, Florentine Codex, A. Anderson and C. Dibble (ed. and trans.), (1970)

(3) M. Leon-Portilla, The Broken Spears: The Aztec Account of the Conquest of Mexico, (1959)

(4) Juan de Betanzos, Narrative of the Incas, R. Hamilton and D. Buhanan (ed. And trans.), (1996)

(5) Garcilaos de la Vega, Royal Commentaries of the Incas and General History of Peru, H. V. Livermore (ed. and trans.), (1987)

(6) Broken Spears, p44

(7) Broken Spears, p51

If you are interested in non-American indigenous peoples, there's an account by a Maori chieftain, Te Horeta te Taniwha, available online, of his memories of first encountering Captain Cook's ship and crew, and the captain himself, as a boy. It was recorded in 1852 so long after Cook's three voyages in the 1770s, on the other hand oral history was/is important amongst Maori.