No, Chinchillas cannot eat roaches.
There are a couple reasons why it isn’t a good idea for a chinchilla to eat a roach.
One thing is that chinchillas like to be vegetarians and only eat veggies. A lot of the time they will not take non-vegetarian foods.
It is better for a chinchilla to each such things as hay, grass and related items as they have more nutritional benefits for chinchillas.
Also roaches are insects that may carry bacteria that could cause harm to a chin so its best for them to avoid eating this insect.
And finally another thing that may occur is that a roach could bite the chinchilla which is not ideal and therefore you should avoid letting your chin eat this insect.