They can appeal, but it is pretty much impossible for them to win unless they have video evidence. The onus is on them to prove innocence, whereas it used to be on the CFA to prove guilt. These days a red card is just a tick box so, unlike before, the club doesn't have a copy of the referee's report, which means they can't base their defence on what the referee has said.
I did hear about one appeal that was successful, where a keeper was sent off for DOGSO for handling outside his area. They appealed on the basis of there being no attacking players close when it happened, and when asked the referee agreed but said that any handling outside the area is an automatic red card. So that was a successful appeal and education for the referee in question.