If a chinchilla can get its head through, which, to me, seems likely, it can extrude the rest of its squishy self through the hole. I think it has escaped. Were any exterior doors to the house left open? If so finding your pet will be highly unlikely. If your dog is not good around animals it may have eaten it. My neighbor had this happen to his. I'm not intentionally trying to upset you but I don't want your hopes to be unreasonable.
If it were my chin I would get some help and do a systematic search closing each door to each room after it has been searched. Also turn off all fans or things that make noise and listen carefully for any noises.