The most immediate use of they is probably because Peter Clemenza has recruited Rocco Lampone to assist with the execution of Paulie Gatto. So the they stands for Clemenza and Lampone, who will both be in the car when the execution of Gatto is, um, executed (carried out/done).
The words that come right before your excerpt are:
And why Lampone? Paulie would find that most alarming, especially since Lampone had to be in the rear seat. Paulie wouldn’t like being helpless at the wheel with Lampone behind his head. Clemenza rubbed and polished the metal of his Cadillac furiously. It was going to be tricky. Very tricky.
Also note that the first sentene of your excerpt refers to the the thought of bringing in a third man ("a second accomplice") to help with the execution. Clemenza decides against this, but there will still be two, Clemenza and Lampone, who are they.
[Edit to include: A much lesser possibility is that by they, Clemenza is referring to the whole Corleone Family. And that by Clemenza--as an agent of the family--sticking to procedure, the family Corleone will be sticking to procedure through him. But I think this stretches the point and ignores the simplest, naturalist explanation.]
Finally, I doubt you will find (m)any uses of they as a genderfree pronoun in The Godfather.
Chapter 6 in full.