Applying to PhD programs - One reference just quit his job

The first thing you need to do is check if your reference is still willing to act as a reference even though they have changed jobs. In general, they will, but if they have retired or left the field completely maybe they do not want to. Potentially, a reference who works in industry has signed some sort of non-compete agreement or an NDA that does not allow them to write a reference any longer. Once you either get the referee's new contact information, or a new reference, you need to email your potential supervisors.

In the email, you need to tell the potential supervisor what happened. A job change is pretty typical and does not look bad in any way. Having to change referees is a little more problematic, but just explain briefly what happened.

I don't see much of the problem here. You are applying to a PhD not a faculty position. The second reference should or might be contacted by email. The fact that he quit his job has no relevance (unless he tries to politely decline serving as a reference for you).

Otherwise, just ask your second reference to email directly the reference to the Professor.

The point being is that the bureaucratic side is unimportant.