If someone comes up to daven as a chazzan, and you have questions about how kosher they are or whether they're truly shomer shabbos, are you allowed to still use him as a chazzan, or are there halachic ramifications?
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If you know the fellow publicly flouts shabbos-observance, then we have a problem. (And what does "publicly flout" mean in today's world is another question; Rabbi Joseph Soloveichik is quoted as saying that if he'd be embarrassed to be seen doing it while the rabbi walked by, that's still not "publicly flouting.")
Rabbi Shternbuch was asked regarding a small community in South Africa which, for High Holidays, had to choose between a cantor who kept a lot but not shabbos; and one who kept everything, but as he was a Kohen and couldn't have his non-Jewish girlfriend convert, married her anyhow. What was worse, the shabbos violator or the married-to-a-non-Jew? Messy dilemma, rabbis debated this one.
But generally, best not to worry about it, unless having this fellow daven will cause a storm, e.g. he openly flouts community standards. Here's Kitzur Shulchan Aruch 128:7, regarding choosing a chazzan for High Holidays (for which the bar is higher):
One more note: with regards to who you call to the Torah, Rabbi Moshe Feinstein felt that the average non-observant Jew is okay (though observant is slightly more preferable).