Why is my chinchilla's mouth and chest wet? Is it a problem?

Thanks for your question, and thank you for the pictures - they are very helpful. When we see wetness around the mouth, chest, and sometimes even the feet, of chinchillas one of the first concerns that we have is that there may be some dental disease present. Chinchilla teeth are always growing (unlike ours) so they need to be eating a lot of fiber (mostly hay) to wear them down. If they don't eat enough fiber, or if their genetics are predisposed to having them develop dental issues, then their teeth grow abnormally, causing points and spurs to form. These can cut up the tongue and cheeks, and are painful, which results in drooling. This would be an important disease to rule out in Izzy. To do this, you will need to take him to a veterinarian, as it is very difficult to see all the way to the back of a chinchilla's mouth at home. They will also feel the jaw bones of Izzy, as often there are changes with the jaws when dental disease is present. Depending on what your vet finds, they may recommend diagnostic imaging, such as radiographs (x-rays) to further evaluate the teeth.
I hope that gives you a place to start, and I wish you and Izzy the best of luck!