Hi Melissa...as a fellow chinchilla lover, I can really empathize with your concern. These little guys are so much fun and so active, it is really depressing to see them when they don't feel well.
What are you feeding right now? As you may already know, chinchillas can not make Vitamin C and it must be supplied in their diet. If for some reason you are using rabbit pellets instead of chin pellets, that could be an issue. I know that this wouldn't make him shake his head, but if his immune system was supressed due to the wrong food, it could make him more susceptible to ear issues and other health problems.
Is his cage near an air conditioning vent? You might check the placement of the cage and make sure he isn't right over or right underneath a vent. Similarly, does he have a "cave" where he can go and feel safe?
I have to concur with Dr. Humphries...an exam by your veterinarian could tell if this was a simple case of some alfalfa stems caught in his ear or a more complex case of an ear infection. As I have often told clients...spendng the money on a physical exam is the first and best step to helping your pet get better. If there is something more severely wrong, then you and your veterinarian can talk about options for treatment and financing at that time.
Could be a simple ear infection, or something caught in the ear. A simple exam would determine this.
I would be sure his housing is warm so he doesn't waste energy on trying to maintain body temps.
While I understand economic concerns, a simple and inexpensive exam is needed and we will hope the treatment is also simple and affordable.