The De Soto Chronicles:The Expedition of Hernando De Soto to North America 1539-1543
Which is a wonderful resource excepting that the PDF file consists of scanned pages and is not a readily searchable database. I have though found on-line OCR scans of the major accounts, which will allow one to to find "crossbow" or "axe' for example, easily in the narrative text.
Luys Hernandez de Biedma was the "factor" or Royal official on the expedition. His is the shortest account and can be called the report of Soto's entrada. Here its presented in Spanish and English side by side.
RELACION DE LA ISLA DE LA FLORIDA /RELATION OF THE CONQUEST OF FLORIDA PRESENTED BY LUYS HERNANDEZ DE BIEDMA IN THE YEAR 1544 TO THE KING OF SPAIN IN COUNCIL
Rodrigo Ranjel was Hernando de Soto's secretary and seems to have kept a diary, though towards the end of the four year long march entries become sparse.
A NARRATIVE OF DE SOTO'S EXPEDITION BASED ON THE DIARY OF RODRIGO RANJEL
The Gentlemen of Elvas, as the unnamed Portuguese adventuer has become known, wrote an account of the expedition.
The above three accounts are the primary accounts that have survived the centuries. Garcilaso's Florida of the Inca is technically considered a secondary work as Garcilaso was not a member of the expedition and his narrative is based largely on the memories of an aging conquistador he knew in Peru. It is by far the largest (Its Vol. II of the Chronicles - Vol I, is Biedma, Ranjel,Elvas, and other material) and most readable of the relations, however its the most embelished in terms of numbers and least reliable in dates and place names.
Another resource I should mention is the work of Buckingham Smith, a nineteenth century translator who includes in an appendix some of the testimonies (usually as part of a request for a royal pension) of some of the ordinary participants on the expedition. No one seems to have gathered together and published this kind of information as the Flint's have done with the Coronado expedition. Among these testimonies is the only mention of Ana Mendez an eleven year old servant girl who accompanied the expedition,
Narratives of the Career of Hernando de Soto in the Conquest of Florida By Buckingham Smith
starting around page 289 on Google Books. Likewise you can also download a zip file of
Avellaneda's Los Sobrevivientes de la Florida: The Survivors of the De Soto Expedition
where in one finds that Soto's tent began to mildew and rot out within a month of landing in Florida. but it doesn't contain the information in a raw translated form.