That would be tricky, since you can't give too many treats, and also it would depend on what you are trying to train. It can be done, I've never bothered to try myself, but I've heard of and see people doing minor tricks like spin around, come, touch a target, or even go back to the cage. However chinchillas don't think the same way as a dog, which tend to do thing to please you, they are more like cats, they don't care too much if they please you they think more along the lines of "why? what is in it for me?" so I'm not sure you would ever be able to space out (go a few clicks between giving a treat) or remove the treats from the equation like you can with a dog. It also means if you don't have a high enough value treat they may not cooperate, and high value treats tend to be the more "junk food" type ones, which you need to ration the most, lol.
I don't use a clicker myself, but I thought one of my chinchilla's (the other one I don't have a good enough bond with yet, still working on that) a couple of things like.
"Come" "Take it"(Whenever I give a stick or something) "Gimme kiss"(He then presses nose against my lips) "Leave it"(Whenever he trys biting something that is not allowed, like cables.. Or trys to roll through his pee pan.....)
This I did by praising a lot, like "Good boy!" Or "Good job!" In a tiny bit of a higher voice than normal and in the beginning I mostly gave rolled oats or rose hips in very tiny pieces, maybe about 1 4th of a rolled oat.
Just try and see how your chinchilla reacts to it, not all will do it and keep in mind that they are very playful, so even after 6 years mine still doesn't always come when I tell him to, because he likes the chase >-< but 90% of the time he will come back after saying it a few times and other times I leave him be and he will come eventually back lol
Hope this helped a bit? (Also reason for not using a clicker is because they can hear a lot better than a dog, yet they are so small I'm pretty sure if it was used that they wouldn't really appreciate the sound)