The first post-contact documents written in a non-European language were most probably Jewish community documents pertaining to weddings and such, which are traditionally written in Hebrew. A small Sephardic Jewish community lived in New Amsterdam from the 1650s onwards. From at least 1695 this community, called Shearith Israel ("remnant of Israel"), had its own synagogue.
The earliest thing that one could call documents are the Dresden, Madrid, and Paris codices from the Post-Classic Maya (~1000-1200). We do have historical evidence for early codices destroyed by the Spanish, and images of scribes writing codices, that date their use to the beginning of the Classic Maya period (~300-400). They are all written using the Mayan hieroglyphic script and mostly describe astronomical, astrological, and calendrical phenomena.
If you extend document to engraved items I would say the Epi-Olmec texts that Kaufman and Justeson have attempted to translate.
hi! you may not be quite what you're looking for, since both originals have been translated and one lost, but you may be interested in these examples:
1880: Abraham Ulrikab's diary is lost, but a German translation survived & now there's an English translation, The Diary of Abraham Ulrikab (2005) edited by Hartmut Lutz. The book got a mention in this thread: Are there any documented cases of Native Americans visiting or living in London or some other city in Europe?
ca 1970: (not old, but it covers life from the turn of the century) Peter Pitseolak's notes have been translated & compiled as People from Our Side: A Life Story with Photographs and Oral Biography, aka People from Our Side: An Inuit Record of Seekooseelak, the Land of the People of Cape Dorset, Baffin Island a Life Story with Photographs (1975) by Peter Pitseolak & Dorothy Harley Eber. The book got a couple of mentions in this sub in Saturday Sources | March 30, 2013 and How did Peter Pitseolak's work affect understanding of Inuit culture by non-Inuit? How did it affect the Inuit's understanding of themselves?