The Medici appear in Florence sometime in the eleventh century, and their main line extinguishes itself with the death of Gian Gastone in 1737. Through six centuries, their contemporaries left a vast amount written accounts, published works, as well as letters and manuscripts.

However, I have the feeling you're asking about Lorenzo de Medici and/or his grandfather Cosimo, who led the family during a particularly colorful time in Italian history. So you're asking if there were any contemporaries who left accounts which historians can consult?

Yes; first and foremost of these are Lorenzo's own. His letters are conserved in the State Archive in Florence, while a few others written to the Sforza and Papacy conserved in Milan and Rome. The letters have been published in sixteen volumes by the Italian Institute of Renaissance Studies in Florence. Less rigorous catalogues of his letters to specific recipients, like the Papal court in Rome and the Sforza court in Milan, have been published since the 19th century (although my favorite and most handy of these catalogues was actually published a little later, in 1910; under the title Delle relazioni private fra la corte Sforzesca di Milano e casa Medici. In any case, how much a renaissance scholar uses these less complete but readily available catalogue is a pretty good indicator of laziness; I think saw it cited in a paper out of Princeton recently and definetely chuckled).

Specifically dealing with accounts meant to be historical as mentioned in the question, there is the very famous history of Florence titled Istorie fiorentine, Nicolò Machiavelli's final written work. A halfhearted attempt by the then-exiled Machiavelli to get into the good graces of the Medici (and written on a shoe-string stipend granted by Pope Clement VII) the text is a rather contorted narrative interpreting history according to Machiavelli's political philosophy while simultaneously trying to exalt the Medici.

A more neutral account could be that of Marco Parenti, whose letters and journals were recently analyzed and consolidated into a narrative. Parenti was a born into a prosperous but unassuming family in the silk merchant's guild, however he quickly abandoned his practice after marrying Caterina Strozzi, a member of an important banking family. Renting out his workshop, Parenti abandoned the silk merchants guild and went to work for his in-laws. An intelligent and hardworking man, Parenti ran the bank's headquarters in Florence while his brothers-in-law travelled around Italy tending to their banking network. Elevated to a number of different political offices in the 1460s and 1470s, Parenti maintained cordial friendships with all of Florentine high society. However, he was very suspicious of the Medici; as a middle-class man who had attained his station through hard work and a positive personality, he was a strong believer of the virtues of the Florentine Republic. The Medici's continuous undermining of the Florentine institutions troubled him deeply.

I could go on with more examples, but it would help to narrow down the focus. Why the interest in primary sources? If you are interested in works about the Medici, why not look for a more modern history? A meticulous author will certainly have used all the sources outlined above and many more. Government of Florence Under the Medici, 1434-94 by good old Nicolai Rubinstein is a time-tested favorite, while the popular British writer (but non-historian) Christopher Hibbert wrote a more recent book titled The House of Medici: Its Rise and Fall.