Is it appropriate to use chapter breaks in works shorter than novel-length? I feel as though the text requires breakpoints peridocially, and if it were a novel, I'd place chapter breaks there, but in a shorter work (!25k words), I'm not sure whether that's appropriate. If not, what should I use?
It is absolutely acceptable. If your story requires a shift in scenes, for example, it would be totally appropriate to indicate that with some form of break. Whether you identify this by using extra space, an image, or any other means is ultimately a matter of personal preference. The main point is that you are using a common device to let your reader know that you are making some sort of shift in your story.
Yes, chapters are appropriate for the most part. If you're writing genre fiction, your editor will almost certainly separate into chapters if you don't, anyway--readers often rely on them.
If you're not writing genre fiction, of course, you can do whatever the heck you like :) but what is important to keep in mind is that novellas are notoriously hard to structure, and chapters (or purely separating into three parts--beginning, middle and end) can aid your construction.