"Degli Arditi Orchestra" is, I think, an ungrammatical or perhaps archaic or dialectical way of writing the band name.
"Orchestra" is the head noun and "degli Arditi" ("of the Arditi," plural) modifies it, so "Orchestra" should be before "degli Arditi." It could be that the name of the band was written the same way in a recording ledger and it was just copied that way on the label. If the intended meaning was "by the Arditi Orchestra" it would say "dell'Arditi Orchestra," "dell'" being singular. Here's another of their discs where the band name is written correctly.
Gnt I-4786 - it is Italian. Degli (like Dell') means "of the" (also "some"), so it seems not to be a part of the credit. Gnt I-4788 is by a singer "Acc. Dell' Orchestra Napoletana". I-4785 is also "Degli Arditi Orchestra" (taken from the blue label images).
I have a small image of 2 pages from the Gennett cat. supplement of Jan 1922, describing new issues 4800/4801, and listing 6 issues (nrs unreadable) of "Italian Records for January", all with I- prefix. The S-prefix means Spanish (e.g., S-4825 Nathan Glantz y su orquesta).