Well if you can't use a tablet, then the way I see it, you've got two options open that I can think of right off the top of my head.
Option One, work traditionally. Draw by hand on paper. Even if you don't have a scanner, there are scanner apps you can get for you phone that will get you a good quality image. Then you can clean it up and upload that.
Option Two, work in vectors. If you use a browser-based program like Inkscape you'll have no problem working on a Chromebook, and vector art can be done easily with a mouse. In fact, I prefer using a mouse when I'm working with vectors. It does require a completely different approach to drawing than what you may be used to, but you can get some very high quality results using vector art once you get used to it.