I don't see why you couldn't do this, as long as you made it clear what you were doing, but I think it would probably be a mistake unless it is in addition to the usual documentation (teaching evaluations, teaching letter from a faculty member).
At the very least, you would have to choose an unusually thoughtful and articulate student, someone who could discuss what's distinctive and valuable about your teaching. Even if you did that, there would be some concern that the student is not someone with a letter writing reputation they have an incentive to protect, or that you may have chosen a student because you couldn't find faculty willing to vouch for your teaching. However, I think there's a more fundamental difficulty.
Any half-decent teacher can find at least one student every few years who really thinks highly of their teaching. Maybe it's because the teaching style is a perfect match for the student; maybe it's because the student deeply loves the class material and is somewhat awestruck by the professor. However, there's always someone, so getting a great recommendation from a student or two tells little about how the other students felt.
If you have broader evidence too, such as strong teaching evaluations from your students overall and a positive letter from a faculty member, then this would not be a difficulty. (However, if you have these things then the student letter would probably not be needed.)