Can a Jew who does not observe Shabbat say kiddush for others?
I attended a halacha shiur when I was in yeshiva that addressed this issue. I believe that the posek that my rav was quoting was Rav Ovadiah Yosef zt"l; however, unfortunately, I lost my notes from that shiur, so I cannot confirm that this is correct.
He said that, in general, one should not hear kiddush from someone who is mechalel shabbat; however, if a kibbud av scenario comes up (i.e. non-religious father wants to make kiddush), one may be lenient in such a scenario as long as the father believes that God exists, even if he does not follow halacha. In this scenario, he should use wine or grape juice that is mevushal.
If someone finds himself in a situation where someone who is mechalel shabbat or who does not believe in God is making kiddush, he can simply say kiddush quietly to himself along with the person making the bracha.