I imagine a chinchilla would have to be trained to do this from a very young age and it would take someone who is very comfortable with the mechanics of working with such tiny harnesses and handling chinchillas. That would exclude me for sure. I believe a harness is generally dangerous for chinchillas and most small animals.
I've tried harnessing cats, a rabbit, and a Congo African Grey parrot in the past and have been very unsuccessful. I've been successful with horses and goats that we used to take to outdoor church functions and as live props in plays so it was not like we were totally clueless. Still even with larger animals there is always a danger of accidents and the animals must be supervised closely at all times.
When we (Mom and I and my brother) tried it with our three cats two of them managed to quickly slip it off no matter how we adjusted it. One cat got her front leg partially through and panicked. This situation would be very dangerous to a chin because of the fragility of their joints. If the chin manages to flail around in a loose or maladjusted harness it could easily tear muscles and tendons. If it slips along the neck while it slips a leg through or gets it caught on something suffocation can quickly occur.
There was also rabbit that we harnessed a few times when we let him roam on the floor. The harness came from a pet store but the design itself proved to be dangerous when it ensnared its neck. We managed to cut it off with a knife. Its eyes morbidly bulged out and he nearly suffocated. It was a miracle he survived and we never tried to harness a rabbit since.
This kind of thing really isn't for the faint of heart!