When he is in negotiations with the Archbishop Michael says that the Corleones are totally out of the casinos and that all their money is legitimate (that's a lie of course because Michael had yet to cash out with the commission) which indicates to me that
all the money they had amassed over the years had been laundered.

Don't forget The Family already had great wealth under Vito Corleone, perhaps $100 million, so it is not a stretch to think they had that kind of cash available.