Is it appropriate for my 15 year old son to share a bed room with a 9 year old daughter?

No, its not appropriate. They should sleep in separate rooms. Also they are at very different stages in life. Its not a good mix at all.

Read your laws for your state it could actually be illegal for them to share a room. I am divorced and my ex was having my 7 year old daughter (at the time) share a room with a 10 year old boy (her half brother) my attorney put a stop to that immediately because child services could have been called on that issue.

First no there are no "state laws" many poor families have to all live in one or two bedroom apartments all the time. I am not aware of any laws in any state.

There are rules or guidelines and if you are going though child custody, one parent could use that against another.
And of course there would have to be a report to CS for them to investigate and there would have to be some other issue besides just sharing, there would have to be a claim of some abuse because of it.

Is it proper no, does one child have to sleep at times on a couch or use twin beds in a room, yes it has to happen but should not if there is any way not to.

Sorry you are correct no state law, however child protection agency may take the children away from the parents if a complaint is filed by neighbor, child, friend or family member. Its best to keep the children separate.