Is it permissible to hold a vocal performance competition where the cash prizes are sourced from the price of tickets paid by the audience, with the participants being exempt from any fees?

There is no objection to a competition for determining the adhan time, as long as the prize money comes from sources other than the competitors, thus eliminating any suspicion of gambling. It is a condition for the permissibility of offering prizes in competitions generally that they be for activities that aid in the propagation and establishment of religion.