How to talk to previous lab members when applying to a postdoc?
Should I let the professor know that I am going to email their previous lab members?

There is no reason to do that.
I suggest it may be more useful to contact people who have left the lab, as they are less likely to feel a conflict of interest when describing their experiences.
Ask about the supervisor's management style.  Ask about how working in the lab advanced peoples' careers.  If you are unfamiliar with the location, ask about difficulties people encountered living there.