I'd doubt that the DUI would be a problem, unless there's a blanket prohibition on any felony conviction at all.  The larceny record is a much larger issue, because that's a crime of dishonesty, and the licensing authority is responsible for seeing that real estate sales people have "good moral character."

You should be prepared to show that you have, since serving your time for the larceny conviction, been a law-abiding citizen and that you have held jobs that establish your trustworthiness.  I can't promise you that doing that will keep your license application from being denied.