You are close, but not quite there.
엘범을 받고 싶어 is grammatically correct and people actually say it. For example, one might say 크리스마스 선물로 앨범을 받고 싶어 (I'd like to get an album for Christmas present). The only thing is that 받다 means "to receive", not to have or own.
For "to have", you can use 갖다 or 가지다 which is its long form. So 앨범을 갖고 싶어 corresponds to "I want to have an album.".
As another exampke, you could say 너 이번 생일에 뭐 가지고 싶니? (What would you like to have for your upcoming birthday?).