Modern yakuza organizations tend to camouflage themselves using common terms used by normal companies and NPOs. The name of such an organization itself can be camouflage. ～組 is the traditional naming convention which is strongly associated with yakuza (and construction companies), but some use confusing names like ～興業. From Wikipedia:
The heads of yakuza organizations used to be called 頭領 or 親分, but in some organizations it's now 組長, 総長 or even 会長. 会長 (≒"chairman") is the word used by most large Japanese companies including Toyota and Sony. So I won't be surprised if a yakuza member calls his organization 会社, although I don't know how common it is. Anyway, this does not mean he believes his organization is a proper company.
In addition, some (most?) yakuza organizations run real companies to make money, legally or illegally. So he may be referring to such a real company.