You can make a hole in the wood to fit the feeder, but, unless you already have the equipment, it might be a bit expensive that way. The equipment I am speaking of in the same that is used to put round holes in doors to install the doornobs. The cutting circles come in different sizes.
Unless the feeding dish is heavy ceramic and has straight sides, it would probably not work very well. It would probably be best just to go to the pet store, in the bird section and get one of the metal cups that set in a holder that is attached to the side of the cage. OR you can get a "J" feeder and install that in the side of the cage. The metal cup in the holder would be the easiest and least expensive for the price. The cups can be removed and cleaned easily.
When you've had chins as long as I have, especially as many as I have, you have to learn all the ways to save money and still have plenty of safe toys and things for your chinchillas.
Vinegar ... You could still use a vinegar and water solution, but you would have to remove them, clean them and let them dry completely before putting them back in the cage. A hand vac and sandpaper usually work pretty good on the wood shelves. The only time you have to be concerned is if they have soft poos and the poos get mashed into the shelf, or if one pees on the shelf .... OR if you have a sick chin ... if you have a sick chin I always suggest sterilizing the cage and everything in the cage and replacing the parts that can not be safely/easily sterilized (such as wood and plastic).
When it comes to cleaning wood and plastic ... if it is liquid and not safe for a human to consume, I don't use it for a chinchilla. Wood and plastic are porous and will retain some of the chemicals used to clean them.
Jo Ann