Pigliare un quadro di sito seems to mean more or less occupy an area.
Luca Landucci writes in 1505 in his Florentine Diary:
E questo è un disegno di fare un tempio a San Giovanni Vangiolista, in quel luogo ove egli è, dirimpetto a San Lorenzo; cioè pigliare un quadro quanto tiene la piazza di San Lorenzo, ch'è circa cento braccia per ogni verso, come per una scritta l'ò avvisato
And this is a plan to build a church to St. John the Evangelist, in that area in front of S. Lorenzo; that is, to take|reserve|occupy an area as wide as S. Lorenzo square, which is around one hundred yards in each direction, as I notified him in writing...
The word sito usually means a location, so that you would say "Nel sito sono stati ritrovate sepolture del 1300" (Burials from 1300 have been found at the site). It is also an old participle for situato, meaning located: "l'immobile sito in Vicolo Stretto..." (The building located in Narrow Alley...). But it also has1 an archaic meaning of space: occupare quanto più sito possibile, occupy as much space as possible.
In that case, sito would have been needed to specify what the quadro was to be taken of: a quadro of meadow, of water, of empty air?, hence quadro di sito.
It is not clear whether this area is to be kept free as to allow passage for the coaches, or whether the procession is to be constrained inside the area so that space enough for the coaches is left around it.
1 As a curiosity, in Tuscan vernacular it has the further meaning of especially bad smell, pong: if you hear senti che sito in Florence, they're not complimenting the site - they're bemoaning the horrible smell (there was even a local newspaper title on the Florence landfill, Case Passerini, making a pun and referring to the site of the landfill with an ambiguous il sito di Case Passerini).